Fault Tolerant Facility Location
Chaitanya Swamy David ShmoysCornell University
Theory Seminar 042002
Metric Facility Location
F set of facilitiesD set of clients
Facility i has facility cost fi
cij distance between any i and j in D F
Client j wants to be connected to rj distinct facilities
3
2
2
client
facility
Theory Seminar 042002
We want to
1) Pick a set S of facilities to open
2) Assign each client j to rj open facilities
Goal Minimize total facility cost of S + sum of distances(service cost)
3
2
2client
facility
openfacility
Theory Seminar 042002
rj=1
bullLP rounding Shmoys Tardos amp Aardal Chudak amp Shmoys Sviridenko
bullPrimal-dual algorithms Jain amp Vazirani Markakis Mahdian Saberi amp Vazirani(MMSV01) Jain Mahdian amp Saberi(JMS02)
bullLocal search Koropulu Plaxton amp Rajaraman Guha amp Khuller Charikar amp Guha
Best approx - Mahdian Ye amp Zhang (MYZ02) 152
Previous Work
Theory Seminar 042002
Previous Work (contd)
Uniform requirements rj=r
bullMarkakis et al (MMSV01) 1861
Non-uniform requirements rj
bullJain amp Vazirani O(log rmax)
bullGuha Meyerson amp Munagala 247
Our Results
bullNon-uniform rj get a 2076-approx
bullrj=r can extend JMS02 MYZ02 to
get a 152-approx
Theory Seminar 042002
LP Formulation
Primal
Min i fiyi + ji cijxij
st
i xij ge rj j
xij le yi i j
yi le 1 i
xij ge 0 yi ge 0 i j
Theory Seminar 042002
LP Formulation (contd)
Max j rjvj - i zi
st
vj le wij + cij i j
j wij le fi + zi i
vj ge 0 wij ge 0 zi ge 0 i j
Dual
Theory Seminar 042002
Complementary Slackness
Primal Slackness Conditions
bullxij gt 0 vj = wij + cij
bullyi gt 0 j wij = fi + zi
Dual Slackness Conditions
bullvj gt 0 j xij = rj
bullwij gt 0 xij = yi
bullzi gt 0 yi = 1
Theory Seminar 042002
4-approximation outline
Basic Idea vj lsquopaysrsquo for each cij stxij gt 0
Bound service cost for each copy of j by ρvj total service cost leρj
rjvj
Problem Have ndashzis in the dual
But zi gt 0 yi = 1 So can open these facilities and charge all of this cost to the LP
2
j(1) j(2)
le vj
le vj
view as rj
copiesj(c) cth copy
Theory Seminar 042002
The Algorithm
Phase 1 Clustering Ensures that each copy j(c) has a nearby open facilityIterative algorithm
S = j|rj gt 0 Fj = i|xij gt 0 in fi order
Start of iteration
1 Pick j with smallest vj
2 Cluster is M Fj with iM yi = rj
2
51
2
j
client in Sfacility in some Fj
Cluster M
Theory Seminar 042002
0
X XX
30
2
j
3 Open rj cheapest facilities in M
4 For k st Fk M connect rj copies to opened facilities Decrease rk set Fk=Fk-M
End of iteration
client in S
facility in some Fj
client not in S
X facility removed from
Fj
Cluster M
facility opened from M
Theory Seminar 042002
Analysis Phase 1
Solution is feasible each j is connected to rj distinct facilities
Lemma Facility cost lei fiyi
Proof Cost of rj cheapest facilities in M lerj (avg cost) = iM fiyi These facilities donrsquot get used again
Theory Seminar 042002
Analysis (contd)
Lemma For any j and c service cost of copy j(c) le3vj
Proof
vk le vj since k was chosen as cluster center
Service cost le vj + 2vk le 3vj
Cluster M
j(c)le vj
le vk
le vk k
Theory Seminar 042002
The Algorithm (contd)
Phase 2 Taking care of ndashzis
1 Open all (unopened) i st yi = 1
2 For any j if xij = yi = 1 disconnect a copy of j and connect it to i
j
rj = 3
X
i with yi = 1i with yi lt
1 and open
Theory Seminar 042002
Analysis Phase 2
Lemma Cost of phase 2 = fi + cij = j ljvj ndash i zi
Proof Each i with zi gt 0 is opened For iL1 all j st wij gt 0 are connected to it So
vj = (service cost) + (fi +
zi)
j ljvj = fi + cij + i zi
Let L1 = i | yi = 1
Lj = i | xij = 1 L1 and lj = |Lj|
iL1 jiLj
j|iLj
jiLjiL1
Theory Seminar 042002
Finally hellip
Theorem Total cost le 4 times the optimal cost
Proof Total cost le
i fiyi + 3j (rj ndash lj)vj + fi + cij
facility cost of phase 1 cost for
copies connected by
phase 1
cost of phase 2
lei fiyi + 3j (rj ndash lj)vj + (j ljvj ndash i
zi )
lei fiyi + 3(j rjvj ndash i zi )
le4OPT
iL1 jiLj
Theory Seminar 042002
A Randomized Algorithm
Idea Open i with probability ρyi
Expected facility cost le ρi fiyi
Hope that each copy j(c) has a nearby facility open and service cost decreases
Not quitehellip no facility may be open
Cluster facilities open ge 1 facility in each cluster
Theory Seminar 042002
Phase 1 Pruning out ndashzis
Open all i st yi = 1
For each j if xij = yi = 1 connect j to i
Let Lj = i | xij = 1 and lj = |Lj|
Cost = j ljvj ndash i zi
Lj
Fj10
rrsquoj = residual reqmt = 6
Lrsquoj
Phase 2
Open all i st frac12 le yi lt 1
For each j let Lrsquoj = i | frac12 le xij lt 1
Connect copies of j to i Lrsquoj
Lose a factor of 2
facilities opened in
phases 1 2yi = 1 frac12 le yi lt 1yi lt frac12
Set L1
Set L2
Theory Seminar 042002
Phase 3
Notation facwt(S j) = iS xij
1 Form clusters Each cluster has facwt ge frac12
2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi
3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =
4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Metric Facility Location
F set of facilitiesD set of clients
Facility i has facility cost fi
cij distance between any i and j in D F
Client j wants to be connected to rj distinct facilities
3
2
2
client
facility
Theory Seminar 042002
We want to
1) Pick a set S of facilities to open
2) Assign each client j to rj open facilities
Goal Minimize total facility cost of S + sum of distances(service cost)
3
2
2client
facility
openfacility
Theory Seminar 042002
rj=1
bullLP rounding Shmoys Tardos amp Aardal Chudak amp Shmoys Sviridenko
bullPrimal-dual algorithms Jain amp Vazirani Markakis Mahdian Saberi amp Vazirani(MMSV01) Jain Mahdian amp Saberi(JMS02)
bullLocal search Koropulu Plaxton amp Rajaraman Guha amp Khuller Charikar amp Guha
Best approx - Mahdian Ye amp Zhang (MYZ02) 152
Previous Work
Theory Seminar 042002
Previous Work (contd)
Uniform requirements rj=r
bullMarkakis et al (MMSV01) 1861
Non-uniform requirements rj
bullJain amp Vazirani O(log rmax)
bullGuha Meyerson amp Munagala 247
Our Results
bullNon-uniform rj get a 2076-approx
bullrj=r can extend JMS02 MYZ02 to
get a 152-approx
Theory Seminar 042002
LP Formulation
Primal
Min i fiyi + ji cijxij
st
i xij ge rj j
xij le yi i j
yi le 1 i
xij ge 0 yi ge 0 i j
Theory Seminar 042002
LP Formulation (contd)
Max j rjvj - i zi
st
vj le wij + cij i j
j wij le fi + zi i
vj ge 0 wij ge 0 zi ge 0 i j
Dual
Theory Seminar 042002
Complementary Slackness
Primal Slackness Conditions
bullxij gt 0 vj = wij + cij
bullyi gt 0 j wij = fi + zi
Dual Slackness Conditions
bullvj gt 0 j xij = rj
bullwij gt 0 xij = yi
bullzi gt 0 yi = 1
Theory Seminar 042002
4-approximation outline
Basic Idea vj lsquopaysrsquo for each cij stxij gt 0
Bound service cost for each copy of j by ρvj total service cost leρj
rjvj
Problem Have ndashzis in the dual
But zi gt 0 yi = 1 So can open these facilities and charge all of this cost to the LP
2
j(1) j(2)
le vj
le vj
view as rj
copiesj(c) cth copy
Theory Seminar 042002
The Algorithm
Phase 1 Clustering Ensures that each copy j(c) has a nearby open facilityIterative algorithm
S = j|rj gt 0 Fj = i|xij gt 0 in fi order
Start of iteration
1 Pick j with smallest vj
2 Cluster is M Fj with iM yi = rj
2
51
2
j
client in Sfacility in some Fj
Cluster M
Theory Seminar 042002
0
X XX
30
2
j
3 Open rj cheapest facilities in M
4 For k st Fk M connect rj copies to opened facilities Decrease rk set Fk=Fk-M
End of iteration
client in S
facility in some Fj
client not in S
X facility removed from
Fj
Cluster M
facility opened from M
Theory Seminar 042002
Analysis Phase 1
Solution is feasible each j is connected to rj distinct facilities
Lemma Facility cost lei fiyi
Proof Cost of rj cheapest facilities in M lerj (avg cost) = iM fiyi These facilities donrsquot get used again
Theory Seminar 042002
Analysis (contd)
Lemma For any j and c service cost of copy j(c) le3vj
Proof
vk le vj since k was chosen as cluster center
Service cost le vj + 2vk le 3vj
Cluster M
j(c)le vj
le vk
le vk k
Theory Seminar 042002
The Algorithm (contd)
Phase 2 Taking care of ndashzis
1 Open all (unopened) i st yi = 1
2 For any j if xij = yi = 1 disconnect a copy of j and connect it to i
j
rj = 3
X
i with yi = 1i with yi lt
1 and open
Theory Seminar 042002
Analysis Phase 2
Lemma Cost of phase 2 = fi + cij = j ljvj ndash i zi
Proof Each i with zi gt 0 is opened For iL1 all j st wij gt 0 are connected to it So
vj = (service cost) + (fi +
zi)
j ljvj = fi + cij + i zi
Let L1 = i | yi = 1
Lj = i | xij = 1 L1 and lj = |Lj|
iL1 jiLj
j|iLj
jiLjiL1
Theory Seminar 042002
Finally hellip
Theorem Total cost le 4 times the optimal cost
Proof Total cost le
i fiyi + 3j (rj ndash lj)vj + fi + cij
facility cost of phase 1 cost for
copies connected by
phase 1
cost of phase 2
lei fiyi + 3j (rj ndash lj)vj + (j ljvj ndash i
zi )
lei fiyi + 3(j rjvj ndash i zi )
le4OPT
iL1 jiLj
Theory Seminar 042002
A Randomized Algorithm
Idea Open i with probability ρyi
Expected facility cost le ρi fiyi
Hope that each copy j(c) has a nearby facility open and service cost decreases
Not quitehellip no facility may be open
Cluster facilities open ge 1 facility in each cluster
Theory Seminar 042002
Phase 1 Pruning out ndashzis
Open all i st yi = 1
For each j if xij = yi = 1 connect j to i
Let Lj = i | xij = 1 and lj = |Lj|
Cost = j ljvj ndash i zi
Lj
Fj10
rrsquoj = residual reqmt = 6
Lrsquoj
Phase 2
Open all i st frac12 le yi lt 1
For each j let Lrsquoj = i | frac12 le xij lt 1
Connect copies of j to i Lrsquoj
Lose a factor of 2
facilities opened in
phases 1 2yi = 1 frac12 le yi lt 1yi lt frac12
Set L1
Set L2
Theory Seminar 042002
Phase 3
Notation facwt(S j) = iS xij
1 Form clusters Each cluster has facwt ge frac12
2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi
3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =
4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
We want to
1) Pick a set S of facilities to open
2) Assign each client j to rj open facilities
Goal Minimize total facility cost of S + sum of distances(service cost)
3
2
2client
facility
openfacility
Theory Seminar 042002
rj=1
bullLP rounding Shmoys Tardos amp Aardal Chudak amp Shmoys Sviridenko
bullPrimal-dual algorithms Jain amp Vazirani Markakis Mahdian Saberi amp Vazirani(MMSV01) Jain Mahdian amp Saberi(JMS02)
bullLocal search Koropulu Plaxton amp Rajaraman Guha amp Khuller Charikar amp Guha
Best approx - Mahdian Ye amp Zhang (MYZ02) 152
Previous Work
Theory Seminar 042002
Previous Work (contd)
Uniform requirements rj=r
bullMarkakis et al (MMSV01) 1861
Non-uniform requirements rj
bullJain amp Vazirani O(log rmax)
bullGuha Meyerson amp Munagala 247
Our Results
bullNon-uniform rj get a 2076-approx
bullrj=r can extend JMS02 MYZ02 to
get a 152-approx
Theory Seminar 042002
LP Formulation
Primal
Min i fiyi + ji cijxij
st
i xij ge rj j
xij le yi i j
yi le 1 i
xij ge 0 yi ge 0 i j
Theory Seminar 042002
LP Formulation (contd)
Max j rjvj - i zi
st
vj le wij + cij i j
j wij le fi + zi i
vj ge 0 wij ge 0 zi ge 0 i j
Dual
Theory Seminar 042002
Complementary Slackness
Primal Slackness Conditions
bullxij gt 0 vj = wij + cij
bullyi gt 0 j wij = fi + zi
Dual Slackness Conditions
bullvj gt 0 j xij = rj
bullwij gt 0 xij = yi
bullzi gt 0 yi = 1
Theory Seminar 042002
4-approximation outline
Basic Idea vj lsquopaysrsquo for each cij stxij gt 0
Bound service cost for each copy of j by ρvj total service cost leρj
rjvj
Problem Have ndashzis in the dual
But zi gt 0 yi = 1 So can open these facilities and charge all of this cost to the LP
2
j(1) j(2)
le vj
le vj
view as rj
copiesj(c) cth copy
Theory Seminar 042002
The Algorithm
Phase 1 Clustering Ensures that each copy j(c) has a nearby open facilityIterative algorithm
S = j|rj gt 0 Fj = i|xij gt 0 in fi order
Start of iteration
1 Pick j with smallest vj
2 Cluster is M Fj with iM yi = rj
2
51
2
j
client in Sfacility in some Fj
Cluster M
Theory Seminar 042002
0
X XX
30
2
j
3 Open rj cheapest facilities in M
4 For k st Fk M connect rj copies to opened facilities Decrease rk set Fk=Fk-M
End of iteration
client in S
facility in some Fj
client not in S
X facility removed from
Fj
Cluster M
facility opened from M
Theory Seminar 042002
Analysis Phase 1
Solution is feasible each j is connected to rj distinct facilities
Lemma Facility cost lei fiyi
Proof Cost of rj cheapest facilities in M lerj (avg cost) = iM fiyi These facilities donrsquot get used again
Theory Seminar 042002
Analysis (contd)
Lemma For any j and c service cost of copy j(c) le3vj
Proof
vk le vj since k was chosen as cluster center
Service cost le vj + 2vk le 3vj
Cluster M
j(c)le vj
le vk
le vk k
Theory Seminar 042002
The Algorithm (contd)
Phase 2 Taking care of ndashzis
1 Open all (unopened) i st yi = 1
2 For any j if xij = yi = 1 disconnect a copy of j and connect it to i
j
rj = 3
X
i with yi = 1i with yi lt
1 and open
Theory Seminar 042002
Analysis Phase 2
Lemma Cost of phase 2 = fi + cij = j ljvj ndash i zi
Proof Each i with zi gt 0 is opened For iL1 all j st wij gt 0 are connected to it So
vj = (service cost) + (fi +
zi)
j ljvj = fi + cij + i zi
Let L1 = i | yi = 1
Lj = i | xij = 1 L1 and lj = |Lj|
iL1 jiLj
j|iLj
jiLjiL1
Theory Seminar 042002
Finally hellip
Theorem Total cost le 4 times the optimal cost
Proof Total cost le
i fiyi + 3j (rj ndash lj)vj + fi + cij
facility cost of phase 1 cost for
copies connected by
phase 1
cost of phase 2
lei fiyi + 3j (rj ndash lj)vj + (j ljvj ndash i
zi )
lei fiyi + 3(j rjvj ndash i zi )
le4OPT
iL1 jiLj
Theory Seminar 042002
A Randomized Algorithm
Idea Open i with probability ρyi
Expected facility cost le ρi fiyi
Hope that each copy j(c) has a nearby facility open and service cost decreases
Not quitehellip no facility may be open
Cluster facilities open ge 1 facility in each cluster
Theory Seminar 042002
Phase 1 Pruning out ndashzis
Open all i st yi = 1
For each j if xij = yi = 1 connect j to i
Let Lj = i | xij = 1 and lj = |Lj|
Cost = j ljvj ndash i zi
Lj
Fj10
rrsquoj = residual reqmt = 6
Lrsquoj
Phase 2
Open all i st frac12 le yi lt 1
For each j let Lrsquoj = i | frac12 le xij lt 1
Connect copies of j to i Lrsquoj
Lose a factor of 2
facilities opened in
phases 1 2yi = 1 frac12 le yi lt 1yi lt frac12
Set L1
Set L2
Theory Seminar 042002
Phase 3
Notation facwt(S j) = iS xij
1 Form clusters Each cluster has facwt ge frac12
2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi
3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =
4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
rj=1
bullLP rounding Shmoys Tardos amp Aardal Chudak amp Shmoys Sviridenko
bullPrimal-dual algorithms Jain amp Vazirani Markakis Mahdian Saberi amp Vazirani(MMSV01) Jain Mahdian amp Saberi(JMS02)
bullLocal search Koropulu Plaxton amp Rajaraman Guha amp Khuller Charikar amp Guha
Best approx - Mahdian Ye amp Zhang (MYZ02) 152
Previous Work
Theory Seminar 042002
Previous Work (contd)
Uniform requirements rj=r
bullMarkakis et al (MMSV01) 1861
Non-uniform requirements rj
bullJain amp Vazirani O(log rmax)
bullGuha Meyerson amp Munagala 247
Our Results
bullNon-uniform rj get a 2076-approx
bullrj=r can extend JMS02 MYZ02 to
get a 152-approx
Theory Seminar 042002
LP Formulation
Primal
Min i fiyi + ji cijxij
st
i xij ge rj j
xij le yi i j
yi le 1 i
xij ge 0 yi ge 0 i j
Theory Seminar 042002
LP Formulation (contd)
Max j rjvj - i zi
st
vj le wij + cij i j
j wij le fi + zi i
vj ge 0 wij ge 0 zi ge 0 i j
Dual
Theory Seminar 042002
Complementary Slackness
Primal Slackness Conditions
bullxij gt 0 vj = wij + cij
bullyi gt 0 j wij = fi + zi
Dual Slackness Conditions
bullvj gt 0 j xij = rj
bullwij gt 0 xij = yi
bullzi gt 0 yi = 1
Theory Seminar 042002
4-approximation outline
Basic Idea vj lsquopaysrsquo for each cij stxij gt 0
Bound service cost for each copy of j by ρvj total service cost leρj
rjvj
Problem Have ndashzis in the dual
But zi gt 0 yi = 1 So can open these facilities and charge all of this cost to the LP
2
j(1) j(2)
le vj
le vj
view as rj
copiesj(c) cth copy
Theory Seminar 042002
The Algorithm
Phase 1 Clustering Ensures that each copy j(c) has a nearby open facilityIterative algorithm
S = j|rj gt 0 Fj = i|xij gt 0 in fi order
Start of iteration
1 Pick j with smallest vj
2 Cluster is M Fj with iM yi = rj
2
51
2
j
client in Sfacility in some Fj
Cluster M
Theory Seminar 042002
0
X XX
30
2
j
3 Open rj cheapest facilities in M
4 For k st Fk M connect rj copies to opened facilities Decrease rk set Fk=Fk-M
End of iteration
client in S
facility in some Fj
client not in S
X facility removed from
Fj
Cluster M
facility opened from M
Theory Seminar 042002
Analysis Phase 1
Solution is feasible each j is connected to rj distinct facilities
Lemma Facility cost lei fiyi
Proof Cost of rj cheapest facilities in M lerj (avg cost) = iM fiyi These facilities donrsquot get used again
Theory Seminar 042002
Analysis (contd)
Lemma For any j and c service cost of copy j(c) le3vj
Proof
vk le vj since k was chosen as cluster center
Service cost le vj + 2vk le 3vj
Cluster M
j(c)le vj
le vk
le vk k
Theory Seminar 042002
The Algorithm (contd)
Phase 2 Taking care of ndashzis
1 Open all (unopened) i st yi = 1
2 For any j if xij = yi = 1 disconnect a copy of j and connect it to i
j
rj = 3
X
i with yi = 1i with yi lt
1 and open
Theory Seminar 042002
Analysis Phase 2
Lemma Cost of phase 2 = fi + cij = j ljvj ndash i zi
Proof Each i with zi gt 0 is opened For iL1 all j st wij gt 0 are connected to it So
vj = (service cost) + (fi +
zi)
j ljvj = fi + cij + i zi
Let L1 = i | yi = 1
Lj = i | xij = 1 L1 and lj = |Lj|
iL1 jiLj
j|iLj
jiLjiL1
Theory Seminar 042002
Finally hellip
Theorem Total cost le 4 times the optimal cost
Proof Total cost le
i fiyi + 3j (rj ndash lj)vj + fi + cij
facility cost of phase 1 cost for
copies connected by
phase 1
cost of phase 2
lei fiyi + 3j (rj ndash lj)vj + (j ljvj ndash i
zi )
lei fiyi + 3(j rjvj ndash i zi )
le4OPT
iL1 jiLj
Theory Seminar 042002
A Randomized Algorithm
Idea Open i with probability ρyi
Expected facility cost le ρi fiyi
Hope that each copy j(c) has a nearby facility open and service cost decreases
Not quitehellip no facility may be open
Cluster facilities open ge 1 facility in each cluster
Theory Seminar 042002
Phase 1 Pruning out ndashzis
Open all i st yi = 1
For each j if xij = yi = 1 connect j to i
Let Lj = i | xij = 1 and lj = |Lj|
Cost = j ljvj ndash i zi
Lj
Fj10
rrsquoj = residual reqmt = 6
Lrsquoj
Phase 2
Open all i st frac12 le yi lt 1
For each j let Lrsquoj = i | frac12 le xij lt 1
Connect copies of j to i Lrsquoj
Lose a factor of 2
facilities opened in
phases 1 2yi = 1 frac12 le yi lt 1yi lt frac12
Set L1
Set L2
Theory Seminar 042002
Phase 3
Notation facwt(S j) = iS xij
1 Form clusters Each cluster has facwt ge frac12
2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi
3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =
4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Previous Work (contd)
Uniform requirements rj=r
bullMarkakis et al (MMSV01) 1861
Non-uniform requirements rj
bullJain amp Vazirani O(log rmax)
bullGuha Meyerson amp Munagala 247
Our Results
bullNon-uniform rj get a 2076-approx
bullrj=r can extend JMS02 MYZ02 to
get a 152-approx
Theory Seminar 042002
LP Formulation
Primal
Min i fiyi + ji cijxij
st
i xij ge rj j
xij le yi i j
yi le 1 i
xij ge 0 yi ge 0 i j
Theory Seminar 042002
LP Formulation (contd)
Max j rjvj - i zi
st
vj le wij + cij i j
j wij le fi + zi i
vj ge 0 wij ge 0 zi ge 0 i j
Dual
Theory Seminar 042002
Complementary Slackness
Primal Slackness Conditions
bullxij gt 0 vj = wij + cij
bullyi gt 0 j wij = fi + zi
Dual Slackness Conditions
bullvj gt 0 j xij = rj
bullwij gt 0 xij = yi
bullzi gt 0 yi = 1
Theory Seminar 042002
4-approximation outline
Basic Idea vj lsquopaysrsquo for each cij stxij gt 0
Bound service cost for each copy of j by ρvj total service cost leρj
rjvj
Problem Have ndashzis in the dual
But zi gt 0 yi = 1 So can open these facilities and charge all of this cost to the LP
2
j(1) j(2)
le vj
le vj
view as rj
copiesj(c) cth copy
Theory Seminar 042002
The Algorithm
Phase 1 Clustering Ensures that each copy j(c) has a nearby open facilityIterative algorithm
S = j|rj gt 0 Fj = i|xij gt 0 in fi order
Start of iteration
1 Pick j with smallest vj
2 Cluster is M Fj with iM yi = rj
2
51
2
j
client in Sfacility in some Fj
Cluster M
Theory Seminar 042002
0
X XX
30
2
j
3 Open rj cheapest facilities in M
4 For k st Fk M connect rj copies to opened facilities Decrease rk set Fk=Fk-M
End of iteration
client in S
facility in some Fj
client not in S
X facility removed from
Fj
Cluster M
facility opened from M
Theory Seminar 042002
Analysis Phase 1
Solution is feasible each j is connected to rj distinct facilities
Lemma Facility cost lei fiyi
Proof Cost of rj cheapest facilities in M lerj (avg cost) = iM fiyi These facilities donrsquot get used again
Theory Seminar 042002
Analysis (contd)
Lemma For any j and c service cost of copy j(c) le3vj
Proof
vk le vj since k was chosen as cluster center
Service cost le vj + 2vk le 3vj
Cluster M
j(c)le vj
le vk
le vk k
Theory Seminar 042002
The Algorithm (contd)
Phase 2 Taking care of ndashzis
1 Open all (unopened) i st yi = 1
2 For any j if xij = yi = 1 disconnect a copy of j and connect it to i
j
rj = 3
X
i with yi = 1i with yi lt
1 and open
Theory Seminar 042002
Analysis Phase 2
Lemma Cost of phase 2 = fi + cij = j ljvj ndash i zi
Proof Each i with zi gt 0 is opened For iL1 all j st wij gt 0 are connected to it So
vj = (service cost) + (fi +
zi)
j ljvj = fi + cij + i zi
Let L1 = i | yi = 1
Lj = i | xij = 1 L1 and lj = |Lj|
iL1 jiLj
j|iLj
jiLjiL1
Theory Seminar 042002
Finally hellip
Theorem Total cost le 4 times the optimal cost
Proof Total cost le
i fiyi + 3j (rj ndash lj)vj + fi + cij
facility cost of phase 1 cost for
copies connected by
phase 1
cost of phase 2
lei fiyi + 3j (rj ndash lj)vj + (j ljvj ndash i
zi )
lei fiyi + 3(j rjvj ndash i zi )
le4OPT
iL1 jiLj
Theory Seminar 042002
A Randomized Algorithm
Idea Open i with probability ρyi
Expected facility cost le ρi fiyi
Hope that each copy j(c) has a nearby facility open and service cost decreases
Not quitehellip no facility may be open
Cluster facilities open ge 1 facility in each cluster
Theory Seminar 042002
Phase 1 Pruning out ndashzis
Open all i st yi = 1
For each j if xij = yi = 1 connect j to i
Let Lj = i | xij = 1 and lj = |Lj|
Cost = j ljvj ndash i zi
Lj
Fj10
rrsquoj = residual reqmt = 6
Lrsquoj
Phase 2
Open all i st frac12 le yi lt 1
For each j let Lrsquoj = i | frac12 le xij lt 1
Connect copies of j to i Lrsquoj
Lose a factor of 2
facilities opened in
phases 1 2yi = 1 frac12 le yi lt 1yi lt frac12
Set L1
Set L2
Theory Seminar 042002
Phase 3
Notation facwt(S j) = iS xij
1 Form clusters Each cluster has facwt ge frac12
2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi
3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =
4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
LP Formulation
Primal
Min i fiyi + ji cijxij
st
i xij ge rj j
xij le yi i j
yi le 1 i
xij ge 0 yi ge 0 i j
Theory Seminar 042002
LP Formulation (contd)
Max j rjvj - i zi
st
vj le wij + cij i j
j wij le fi + zi i
vj ge 0 wij ge 0 zi ge 0 i j
Dual
Theory Seminar 042002
Complementary Slackness
Primal Slackness Conditions
bullxij gt 0 vj = wij + cij
bullyi gt 0 j wij = fi + zi
Dual Slackness Conditions
bullvj gt 0 j xij = rj
bullwij gt 0 xij = yi
bullzi gt 0 yi = 1
Theory Seminar 042002
4-approximation outline
Basic Idea vj lsquopaysrsquo for each cij stxij gt 0
Bound service cost for each copy of j by ρvj total service cost leρj
rjvj
Problem Have ndashzis in the dual
But zi gt 0 yi = 1 So can open these facilities and charge all of this cost to the LP
2
j(1) j(2)
le vj
le vj
view as rj
copiesj(c) cth copy
Theory Seminar 042002
The Algorithm
Phase 1 Clustering Ensures that each copy j(c) has a nearby open facilityIterative algorithm
S = j|rj gt 0 Fj = i|xij gt 0 in fi order
Start of iteration
1 Pick j with smallest vj
2 Cluster is M Fj with iM yi = rj
2
51
2
j
client in Sfacility in some Fj
Cluster M
Theory Seminar 042002
0
X XX
30
2
j
3 Open rj cheapest facilities in M
4 For k st Fk M connect rj copies to opened facilities Decrease rk set Fk=Fk-M
End of iteration
client in S
facility in some Fj
client not in S
X facility removed from
Fj
Cluster M
facility opened from M
Theory Seminar 042002
Analysis Phase 1
Solution is feasible each j is connected to rj distinct facilities
Lemma Facility cost lei fiyi
Proof Cost of rj cheapest facilities in M lerj (avg cost) = iM fiyi These facilities donrsquot get used again
Theory Seminar 042002
Analysis (contd)
Lemma For any j and c service cost of copy j(c) le3vj
Proof
vk le vj since k was chosen as cluster center
Service cost le vj + 2vk le 3vj
Cluster M
j(c)le vj
le vk
le vk k
Theory Seminar 042002
The Algorithm (contd)
Phase 2 Taking care of ndashzis
1 Open all (unopened) i st yi = 1
2 For any j if xij = yi = 1 disconnect a copy of j and connect it to i
j
rj = 3
X
i with yi = 1i with yi lt
1 and open
Theory Seminar 042002
Analysis Phase 2
Lemma Cost of phase 2 = fi + cij = j ljvj ndash i zi
Proof Each i with zi gt 0 is opened For iL1 all j st wij gt 0 are connected to it So
vj = (service cost) + (fi +
zi)
j ljvj = fi + cij + i zi
Let L1 = i | yi = 1
Lj = i | xij = 1 L1 and lj = |Lj|
iL1 jiLj
j|iLj
jiLjiL1
Theory Seminar 042002
Finally hellip
Theorem Total cost le 4 times the optimal cost
Proof Total cost le
i fiyi + 3j (rj ndash lj)vj + fi + cij
facility cost of phase 1 cost for
copies connected by
phase 1
cost of phase 2
lei fiyi + 3j (rj ndash lj)vj + (j ljvj ndash i
zi )
lei fiyi + 3(j rjvj ndash i zi )
le4OPT
iL1 jiLj
Theory Seminar 042002
A Randomized Algorithm
Idea Open i with probability ρyi
Expected facility cost le ρi fiyi
Hope that each copy j(c) has a nearby facility open and service cost decreases
Not quitehellip no facility may be open
Cluster facilities open ge 1 facility in each cluster
Theory Seminar 042002
Phase 1 Pruning out ndashzis
Open all i st yi = 1
For each j if xij = yi = 1 connect j to i
Let Lj = i | xij = 1 and lj = |Lj|
Cost = j ljvj ndash i zi
Lj
Fj10
rrsquoj = residual reqmt = 6
Lrsquoj
Phase 2
Open all i st frac12 le yi lt 1
For each j let Lrsquoj = i | frac12 le xij lt 1
Connect copies of j to i Lrsquoj
Lose a factor of 2
facilities opened in
phases 1 2yi = 1 frac12 le yi lt 1yi lt frac12
Set L1
Set L2
Theory Seminar 042002
Phase 3
Notation facwt(S j) = iS xij
1 Form clusters Each cluster has facwt ge frac12
2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi
3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =
4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
LP Formulation (contd)
Max j rjvj - i zi
st
vj le wij + cij i j
j wij le fi + zi i
vj ge 0 wij ge 0 zi ge 0 i j
Dual
Theory Seminar 042002
Complementary Slackness
Primal Slackness Conditions
bullxij gt 0 vj = wij + cij
bullyi gt 0 j wij = fi + zi
Dual Slackness Conditions
bullvj gt 0 j xij = rj
bullwij gt 0 xij = yi
bullzi gt 0 yi = 1
Theory Seminar 042002
4-approximation outline
Basic Idea vj lsquopaysrsquo for each cij stxij gt 0
Bound service cost for each copy of j by ρvj total service cost leρj
rjvj
Problem Have ndashzis in the dual
But zi gt 0 yi = 1 So can open these facilities and charge all of this cost to the LP
2
j(1) j(2)
le vj
le vj
view as rj
copiesj(c) cth copy
Theory Seminar 042002
The Algorithm
Phase 1 Clustering Ensures that each copy j(c) has a nearby open facilityIterative algorithm
S = j|rj gt 0 Fj = i|xij gt 0 in fi order
Start of iteration
1 Pick j with smallest vj
2 Cluster is M Fj with iM yi = rj
2
51
2
j
client in Sfacility in some Fj
Cluster M
Theory Seminar 042002
0
X XX
30
2
j
3 Open rj cheapest facilities in M
4 For k st Fk M connect rj copies to opened facilities Decrease rk set Fk=Fk-M
End of iteration
client in S
facility in some Fj
client not in S
X facility removed from
Fj
Cluster M
facility opened from M
Theory Seminar 042002
Analysis Phase 1
Solution is feasible each j is connected to rj distinct facilities
Lemma Facility cost lei fiyi
Proof Cost of rj cheapest facilities in M lerj (avg cost) = iM fiyi These facilities donrsquot get used again
Theory Seminar 042002
Analysis (contd)
Lemma For any j and c service cost of copy j(c) le3vj
Proof
vk le vj since k was chosen as cluster center
Service cost le vj + 2vk le 3vj
Cluster M
j(c)le vj
le vk
le vk k
Theory Seminar 042002
The Algorithm (contd)
Phase 2 Taking care of ndashzis
1 Open all (unopened) i st yi = 1
2 For any j if xij = yi = 1 disconnect a copy of j and connect it to i
j
rj = 3
X
i with yi = 1i with yi lt
1 and open
Theory Seminar 042002
Analysis Phase 2
Lemma Cost of phase 2 = fi + cij = j ljvj ndash i zi
Proof Each i with zi gt 0 is opened For iL1 all j st wij gt 0 are connected to it So
vj = (service cost) + (fi +
zi)
j ljvj = fi + cij + i zi
Let L1 = i | yi = 1
Lj = i | xij = 1 L1 and lj = |Lj|
iL1 jiLj
j|iLj
jiLjiL1
Theory Seminar 042002
Finally hellip
Theorem Total cost le 4 times the optimal cost
Proof Total cost le
i fiyi + 3j (rj ndash lj)vj + fi + cij
facility cost of phase 1 cost for
copies connected by
phase 1
cost of phase 2
lei fiyi + 3j (rj ndash lj)vj + (j ljvj ndash i
zi )
lei fiyi + 3(j rjvj ndash i zi )
le4OPT
iL1 jiLj
Theory Seminar 042002
A Randomized Algorithm
Idea Open i with probability ρyi
Expected facility cost le ρi fiyi
Hope that each copy j(c) has a nearby facility open and service cost decreases
Not quitehellip no facility may be open
Cluster facilities open ge 1 facility in each cluster
Theory Seminar 042002
Phase 1 Pruning out ndashzis
Open all i st yi = 1
For each j if xij = yi = 1 connect j to i
Let Lj = i | xij = 1 and lj = |Lj|
Cost = j ljvj ndash i zi
Lj
Fj10
rrsquoj = residual reqmt = 6
Lrsquoj
Phase 2
Open all i st frac12 le yi lt 1
For each j let Lrsquoj = i | frac12 le xij lt 1
Connect copies of j to i Lrsquoj
Lose a factor of 2
facilities opened in
phases 1 2yi = 1 frac12 le yi lt 1yi lt frac12
Set L1
Set L2
Theory Seminar 042002
Phase 3
Notation facwt(S j) = iS xij
1 Form clusters Each cluster has facwt ge frac12
2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi
3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =
4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Complementary Slackness
Primal Slackness Conditions
bullxij gt 0 vj = wij + cij
bullyi gt 0 j wij = fi + zi
Dual Slackness Conditions
bullvj gt 0 j xij = rj
bullwij gt 0 xij = yi
bullzi gt 0 yi = 1
Theory Seminar 042002
4-approximation outline
Basic Idea vj lsquopaysrsquo for each cij stxij gt 0
Bound service cost for each copy of j by ρvj total service cost leρj
rjvj
Problem Have ndashzis in the dual
But zi gt 0 yi = 1 So can open these facilities and charge all of this cost to the LP
2
j(1) j(2)
le vj
le vj
view as rj
copiesj(c) cth copy
Theory Seminar 042002
The Algorithm
Phase 1 Clustering Ensures that each copy j(c) has a nearby open facilityIterative algorithm
S = j|rj gt 0 Fj = i|xij gt 0 in fi order
Start of iteration
1 Pick j with smallest vj
2 Cluster is M Fj with iM yi = rj
2
51
2
j
client in Sfacility in some Fj
Cluster M
Theory Seminar 042002
0
X XX
30
2
j
3 Open rj cheapest facilities in M
4 For k st Fk M connect rj copies to opened facilities Decrease rk set Fk=Fk-M
End of iteration
client in S
facility in some Fj
client not in S
X facility removed from
Fj
Cluster M
facility opened from M
Theory Seminar 042002
Analysis Phase 1
Solution is feasible each j is connected to rj distinct facilities
Lemma Facility cost lei fiyi
Proof Cost of rj cheapest facilities in M lerj (avg cost) = iM fiyi These facilities donrsquot get used again
Theory Seminar 042002
Analysis (contd)
Lemma For any j and c service cost of copy j(c) le3vj
Proof
vk le vj since k was chosen as cluster center
Service cost le vj + 2vk le 3vj
Cluster M
j(c)le vj
le vk
le vk k
Theory Seminar 042002
The Algorithm (contd)
Phase 2 Taking care of ndashzis
1 Open all (unopened) i st yi = 1
2 For any j if xij = yi = 1 disconnect a copy of j and connect it to i
j
rj = 3
X
i with yi = 1i with yi lt
1 and open
Theory Seminar 042002
Analysis Phase 2
Lemma Cost of phase 2 = fi + cij = j ljvj ndash i zi
Proof Each i with zi gt 0 is opened For iL1 all j st wij gt 0 are connected to it So
vj = (service cost) + (fi +
zi)
j ljvj = fi + cij + i zi
Let L1 = i | yi = 1
Lj = i | xij = 1 L1 and lj = |Lj|
iL1 jiLj
j|iLj
jiLjiL1
Theory Seminar 042002
Finally hellip
Theorem Total cost le 4 times the optimal cost
Proof Total cost le
i fiyi + 3j (rj ndash lj)vj + fi + cij
facility cost of phase 1 cost for
copies connected by
phase 1
cost of phase 2
lei fiyi + 3j (rj ndash lj)vj + (j ljvj ndash i
zi )
lei fiyi + 3(j rjvj ndash i zi )
le4OPT
iL1 jiLj
Theory Seminar 042002
A Randomized Algorithm
Idea Open i with probability ρyi
Expected facility cost le ρi fiyi
Hope that each copy j(c) has a nearby facility open and service cost decreases
Not quitehellip no facility may be open
Cluster facilities open ge 1 facility in each cluster
Theory Seminar 042002
Phase 1 Pruning out ndashzis
Open all i st yi = 1
For each j if xij = yi = 1 connect j to i
Let Lj = i | xij = 1 and lj = |Lj|
Cost = j ljvj ndash i zi
Lj
Fj10
rrsquoj = residual reqmt = 6
Lrsquoj
Phase 2
Open all i st frac12 le yi lt 1
For each j let Lrsquoj = i | frac12 le xij lt 1
Connect copies of j to i Lrsquoj
Lose a factor of 2
facilities opened in
phases 1 2yi = 1 frac12 le yi lt 1yi lt frac12
Set L1
Set L2
Theory Seminar 042002
Phase 3
Notation facwt(S j) = iS xij
1 Form clusters Each cluster has facwt ge frac12
2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi
3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =
4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
4-approximation outline
Basic Idea vj lsquopaysrsquo for each cij stxij gt 0
Bound service cost for each copy of j by ρvj total service cost leρj
rjvj
Problem Have ndashzis in the dual
But zi gt 0 yi = 1 So can open these facilities and charge all of this cost to the LP
2
j(1) j(2)
le vj
le vj
view as rj
copiesj(c) cth copy
Theory Seminar 042002
The Algorithm
Phase 1 Clustering Ensures that each copy j(c) has a nearby open facilityIterative algorithm
S = j|rj gt 0 Fj = i|xij gt 0 in fi order
Start of iteration
1 Pick j with smallest vj
2 Cluster is M Fj with iM yi = rj
2
51
2
j
client in Sfacility in some Fj
Cluster M
Theory Seminar 042002
0
X XX
30
2
j
3 Open rj cheapest facilities in M
4 For k st Fk M connect rj copies to opened facilities Decrease rk set Fk=Fk-M
End of iteration
client in S
facility in some Fj
client not in S
X facility removed from
Fj
Cluster M
facility opened from M
Theory Seminar 042002
Analysis Phase 1
Solution is feasible each j is connected to rj distinct facilities
Lemma Facility cost lei fiyi
Proof Cost of rj cheapest facilities in M lerj (avg cost) = iM fiyi These facilities donrsquot get used again
Theory Seminar 042002
Analysis (contd)
Lemma For any j and c service cost of copy j(c) le3vj
Proof
vk le vj since k was chosen as cluster center
Service cost le vj + 2vk le 3vj
Cluster M
j(c)le vj
le vk
le vk k
Theory Seminar 042002
The Algorithm (contd)
Phase 2 Taking care of ndashzis
1 Open all (unopened) i st yi = 1
2 For any j if xij = yi = 1 disconnect a copy of j and connect it to i
j
rj = 3
X
i with yi = 1i with yi lt
1 and open
Theory Seminar 042002
Analysis Phase 2
Lemma Cost of phase 2 = fi + cij = j ljvj ndash i zi
Proof Each i with zi gt 0 is opened For iL1 all j st wij gt 0 are connected to it So
vj = (service cost) + (fi +
zi)
j ljvj = fi + cij + i zi
Let L1 = i | yi = 1
Lj = i | xij = 1 L1 and lj = |Lj|
iL1 jiLj
j|iLj
jiLjiL1
Theory Seminar 042002
Finally hellip
Theorem Total cost le 4 times the optimal cost
Proof Total cost le
i fiyi + 3j (rj ndash lj)vj + fi + cij
facility cost of phase 1 cost for
copies connected by
phase 1
cost of phase 2
lei fiyi + 3j (rj ndash lj)vj + (j ljvj ndash i
zi )
lei fiyi + 3(j rjvj ndash i zi )
le4OPT
iL1 jiLj
Theory Seminar 042002
A Randomized Algorithm
Idea Open i with probability ρyi
Expected facility cost le ρi fiyi
Hope that each copy j(c) has a nearby facility open and service cost decreases
Not quitehellip no facility may be open
Cluster facilities open ge 1 facility in each cluster
Theory Seminar 042002
Phase 1 Pruning out ndashzis
Open all i st yi = 1
For each j if xij = yi = 1 connect j to i
Let Lj = i | xij = 1 and lj = |Lj|
Cost = j ljvj ndash i zi
Lj
Fj10
rrsquoj = residual reqmt = 6
Lrsquoj
Phase 2
Open all i st frac12 le yi lt 1
For each j let Lrsquoj = i | frac12 le xij lt 1
Connect copies of j to i Lrsquoj
Lose a factor of 2
facilities opened in
phases 1 2yi = 1 frac12 le yi lt 1yi lt frac12
Set L1
Set L2
Theory Seminar 042002
Phase 3
Notation facwt(S j) = iS xij
1 Form clusters Each cluster has facwt ge frac12
2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi
3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =
4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
The Algorithm
Phase 1 Clustering Ensures that each copy j(c) has a nearby open facilityIterative algorithm
S = j|rj gt 0 Fj = i|xij gt 0 in fi order
Start of iteration
1 Pick j with smallest vj
2 Cluster is M Fj with iM yi = rj
2
51
2
j
client in Sfacility in some Fj
Cluster M
Theory Seminar 042002
0
X XX
30
2
j
3 Open rj cheapest facilities in M
4 For k st Fk M connect rj copies to opened facilities Decrease rk set Fk=Fk-M
End of iteration
client in S
facility in some Fj
client not in S
X facility removed from
Fj
Cluster M
facility opened from M
Theory Seminar 042002
Analysis Phase 1
Solution is feasible each j is connected to rj distinct facilities
Lemma Facility cost lei fiyi
Proof Cost of rj cheapest facilities in M lerj (avg cost) = iM fiyi These facilities donrsquot get used again
Theory Seminar 042002
Analysis (contd)
Lemma For any j and c service cost of copy j(c) le3vj
Proof
vk le vj since k was chosen as cluster center
Service cost le vj + 2vk le 3vj
Cluster M
j(c)le vj
le vk
le vk k
Theory Seminar 042002
The Algorithm (contd)
Phase 2 Taking care of ndashzis
1 Open all (unopened) i st yi = 1
2 For any j if xij = yi = 1 disconnect a copy of j and connect it to i
j
rj = 3
X
i with yi = 1i with yi lt
1 and open
Theory Seminar 042002
Analysis Phase 2
Lemma Cost of phase 2 = fi + cij = j ljvj ndash i zi
Proof Each i with zi gt 0 is opened For iL1 all j st wij gt 0 are connected to it So
vj = (service cost) + (fi +
zi)
j ljvj = fi + cij + i zi
Let L1 = i | yi = 1
Lj = i | xij = 1 L1 and lj = |Lj|
iL1 jiLj
j|iLj
jiLjiL1
Theory Seminar 042002
Finally hellip
Theorem Total cost le 4 times the optimal cost
Proof Total cost le
i fiyi + 3j (rj ndash lj)vj + fi + cij
facility cost of phase 1 cost for
copies connected by
phase 1
cost of phase 2
lei fiyi + 3j (rj ndash lj)vj + (j ljvj ndash i
zi )
lei fiyi + 3(j rjvj ndash i zi )
le4OPT
iL1 jiLj
Theory Seminar 042002
A Randomized Algorithm
Idea Open i with probability ρyi
Expected facility cost le ρi fiyi
Hope that each copy j(c) has a nearby facility open and service cost decreases
Not quitehellip no facility may be open
Cluster facilities open ge 1 facility in each cluster
Theory Seminar 042002
Phase 1 Pruning out ndashzis
Open all i st yi = 1
For each j if xij = yi = 1 connect j to i
Let Lj = i | xij = 1 and lj = |Lj|
Cost = j ljvj ndash i zi
Lj
Fj10
rrsquoj = residual reqmt = 6
Lrsquoj
Phase 2
Open all i st frac12 le yi lt 1
For each j let Lrsquoj = i | frac12 le xij lt 1
Connect copies of j to i Lrsquoj
Lose a factor of 2
facilities opened in
phases 1 2yi = 1 frac12 le yi lt 1yi lt frac12
Set L1
Set L2
Theory Seminar 042002
Phase 3
Notation facwt(S j) = iS xij
1 Form clusters Each cluster has facwt ge frac12
2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi
3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =
4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
0
X XX
30
2
j
3 Open rj cheapest facilities in M
4 For k st Fk M connect rj copies to opened facilities Decrease rk set Fk=Fk-M
End of iteration
client in S
facility in some Fj
client not in S
X facility removed from
Fj
Cluster M
facility opened from M
Theory Seminar 042002
Analysis Phase 1
Solution is feasible each j is connected to rj distinct facilities
Lemma Facility cost lei fiyi
Proof Cost of rj cheapest facilities in M lerj (avg cost) = iM fiyi These facilities donrsquot get used again
Theory Seminar 042002
Analysis (contd)
Lemma For any j and c service cost of copy j(c) le3vj
Proof
vk le vj since k was chosen as cluster center
Service cost le vj + 2vk le 3vj
Cluster M
j(c)le vj
le vk
le vk k
Theory Seminar 042002
The Algorithm (contd)
Phase 2 Taking care of ndashzis
1 Open all (unopened) i st yi = 1
2 For any j if xij = yi = 1 disconnect a copy of j and connect it to i
j
rj = 3
X
i with yi = 1i with yi lt
1 and open
Theory Seminar 042002
Analysis Phase 2
Lemma Cost of phase 2 = fi + cij = j ljvj ndash i zi
Proof Each i with zi gt 0 is opened For iL1 all j st wij gt 0 are connected to it So
vj = (service cost) + (fi +
zi)
j ljvj = fi + cij + i zi
Let L1 = i | yi = 1
Lj = i | xij = 1 L1 and lj = |Lj|
iL1 jiLj
j|iLj
jiLjiL1
Theory Seminar 042002
Finally hellip
Theorem Total cost le 4 times the optimal cost
Proof Total cost le
i fiyi + 3j (rj ndash lj)vj + fi + cij
facility cost of phase 1 cost for
copies connected by
phase 1
cost of phase 2
lei fiyi + 3j (rj ndash lj)vj + (j ljvj ndash i
zi )
lei fiyi + 3(j rjvj ndash i zi )
le4OPT
iL1 jiLj
Theory Seminar 042002
A Randomized Algorithm
Idea Open i with probability ρyi
Expected facility cost le ρi fiyi
Hope that each copy j(c) has a nearby facility open and service cost decreases
Not quitehellip no facility may be open
Cluster facilities open ge 1 facility in each cluster
Theory Seminar 042002
Phase 1 Pruning out ndashzis
Open all i st yi = 1
For each j if xij = yi = 1 connect j to i
Let Lj = i | xij = 1 and lj = |Lj|
Cost = j ljvj ndash i zi
Lj
Fj10
rrsquoj = residual reqmt = 6
Lrsquoj
Phase 2
Open all i st frac12 le yi lt 1
For each j let Lrsquoj = i | frac12 le xij lt 1
Connect copies of j to i Lrsquoj
Lose a factor of 2
facilities opened in
phases 1 2yi = 1 frac12 le yi lt 1yi lt frac12
Set L1
Set L2
Theory Seminar 042002
Phase 3
Notation facwt(S j) = iS xij
1 Form clusters Each cluster has facwt ge frac12
2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi
3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =
4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Analysis Phase 1
Solution is feasible each j is connected to rj distinct facilities
Lemma Facility cost lei fiyi
Proof Cost of rj cheapest facilities in M lerj (avg cost) = iM fiyi These facilities donrsquot get used again
Theory Seminar 042002
Analysis (contd)
Lemma For any j and c service cost of copy j(c) le3vj
Proof
vk le vj since k was chosen as cluster center
Service cost le vj + 2vk le 3vj
Cluster M
j(c)le vj
le vk
le vk k
Theory Seminar 042002
The Algorithm (contd)
Phase 2 Taking care of ndashzis
1 Open all (unopened) i st yi = 1
2 For any j if xij = yi = 1 disconnect a copy of j and connect it to i
j
rj = 3
X
i with yi = 1i with yi lt
1 and open
Theory Seminar 042002
Analysis Phase 2
Lemma Cost of phase 2 = fi + cij = j ljvj ndash i zi
Proof Each i with zi gt 0 is opened For iL1 all j st wij gt 0 are connected to it So
vj = (service cost) + (fi +
zi)
j ljvj = fi + cij + i zi
Let L1 = i | yi = 1
Lj = i | xij = 1 L1 and lj = |Lj|
iL1 jiLj
j|iLj
jiLjiL1
Theory Seminar 042002
Finally hellip
Theorem Total cost le 4 times the optimal cost
Proof Total cost le
i fiyi + 3j (rj ndash lj)vj + fi + cij
facility cost of phase 1 cost for
copies connected by
phase 1
cost of phase 2
lei fiyi + 3j (rj ndash lj)vj + (j ljvj ndash i
zi )
lei fiyi + 3(j rjvj ndash i zi )
le4OPT
iL1 jiLj
Theory Seminar 042002
A Randomized Algorithm
Idea Open i with probability ρyi
Expected facility cost le ρi fiyi
Hope that each copy j(c) has a nearby facility open and service cost decreases
Not quitehellip no facility may be open
Cluster facilities open ge 1 facility in each cluster
Theory Seminar 042002
Phase 1 Pruning out ndashzis
Open all i st yi = 1
For each j if xij = yi = 1 connect j to i
Let Lj = i | xij = 1 and lj = |Lj|
Cost = j ljvj ndash i zi
Lj
Fj10
rrsquoj = residual reqmt = 6
Lrsquoj
Phase 2
Open all i st frac12 le yi lt 1
For each j let Lrsquoj = i | frac12 le xij lt 1
Connect copies of j to i Lrsquoj
Lose a factor of 2
facilities opened in
phases 1 2yi = 1 frac12 le yi lt 1yi lt frac12
Set L1
Set L2
Theory Seminar 042002
Phase 3
Notation facwt(S j) = iS xij
1 Form clusters Each cluster has facwt ge frac12
2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi
3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =
4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Analysis (contd)
Lemma For any j and c service cost of copy j(c) le3vj
Proof
vk le vj since k was chosen as cluster center
Service cost le vj + 2vk le 3vj
Cluster M
j(c)le vj
le vk
le vk k
Theory Seminar 042002
The Algorithm (contd)
Phase 2 Taking care of ndashzis
1 Open all (unopened) i st yi = 1
2 For any j if xij = yi = 1 disconnect a copy of j and connect it to i
j
rj = 3
X
i with yi = 1i with yi lt
1 and open
Theory Seminar 042002
Analysis Phase 2
Lemma Cost of phase 2 = fi + cij = j ljvj ndash i zi
Proof Each i with zi gt 0 is opened For iL1 all j st wij gt 0 are connected to it So
vj = (service cost) + (fi +
zi)
j ljvj = fi + cij + i zi
Let L1 = i | yi = 1
Lj = i | xij = 1 L1 and lj = |Lj|
iL1 jiLj
j|iLj
jiLjiL1
Theory Seminar 042002
Finally hellip
Theorem Total cost le 4 times the optimal cost
Proof Total cost le
i fiyi + 3j (rj ndash lj)vj + fi + cij
facility cost of phase 1 cost for
copies connected by
phase 1
cost of phase 2
lei fiyi + 3j (rj ndash lj)vj + (j ljvj ndash i
zi )
lei fiyi + 3(j rjvj ndash i zi )
le4OPT
iL1 jiLj
Theory Seminar 042002
A Randomized Algorithm
Idea Open i with probability ρyi
Expected facility cost le ρi fiyi
Hope that each copy j(c) has a nearby facility open and service cost decreases
Not quitehellip no facility may be open
Cluster facilities open ge 1 facility in each cluster
Theory Seminar 042002
Phase 1 Pruning out ndashzis
Open all i st yi = 1
For each j if xij = yi = 1 connect j to i
Let Lj = i | xij = 1 and lj = |Lj|
Cost = j ljvj ndash i zi
Lj
Fj10
rrsquoj = residual reqmt = 6
Lrsquoj
Phase 2
Open all i st frac12 le yi lt 1
For each j let Lrsquoj = i | frac12 le xij lt 1
Connect copies of j to i Lrsquoj
Lose a factor of 2
facilities opened in
phases 1 2yi = 1 frac12 le yi lt 1yi lt frac12
Set L1
Set L2
Theory Seminar 042002
Phase 3
Notation facwt(S j) = iS xij
1 Form clusters Each cluster has facwt ge frac12
2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi
3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =
4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
The Algorithm (contd)
Phase 2 Taking care of ndashzis
1 Open all (unopened) i st yi = 1
2 For any j if xij = yi = 1 disconnect a copy of j and connect it to i
j
rj = 3
X
i with yi = 1i with yi lt
1 and open
Theory Seminar 042002
Analysis Phase 2
Lemma Cost of phase 2 = fi + cij = j ljvj ndash i zi
Proof Each i with zi gt 0 is opened For iL1 all j st wij gt 0 are connected to it So
vj = (service cost) + (fi +
zi)
j ljvj = fi + cij + i zi
Let L1 = i | yi = 1
Lj = i | xij = 1 L1 and lj = |Lj|
iL1 jiLj
j|iLj
jiLjiL1
Theory Seminar 042002
Finally hellip
Theorem Total cost le 4 times the optimal cost
Proof Total cost le
i fiyi + 3j (rj ndash lj)vj + fi + cij
facility cost of phase 1 cost for
copies connected by
phase 1
cost of phase 2
lei fiyi + 3j (rj ndash lj)vj + (j ljvj ndash i
zi )
lei fiyi + 3(j rjvj ndash i zi )
le4OPT
iL1 jiLj
Theory Seminar 042002
A Randomized Algorithm
Idea Open i with probability ρyi
Expected facility cost le ρi fiyi
Hope that each copy j(c) has a nearby facility open and service cost decreases
Not quitehellip no facility may be open
Cluster facilities open ge 1 facility in each cluster
Theory Seminar 042002
Phase 1 Pruning out ndashzis
Open all i st yi = 1
For each j if xij = yi = 1 connect j to i
Let Lj = i | xij = 1 and lj = |Lj|
Cost = j ljvj ndash i zi
Lj
Fj10
rrsquoj = residual reqmt = 6
Lrsquoj
Phase 2
Open all i st frac12 le yi lt 1
For each j let Lrsquoj = i | frac12 le xij lt 1
Connect copies of j to i Lrsquoj
Lose a factor of 2
facilities opened in
phases 1 2yi = 1 frac12 le yi lt 1yi lt frac12
Set L1
Set L2
Theory Seminar 042002
Phase 3
Notation facwt(S j) = iS xij
1 Form clusters Each cluster has facwt ge frac12
2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi
3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =
4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Analysis Phase 2
Lemma Cost of phase 2 = fi + cij = j ljvj ndash i zi
Proof Each i with zi gt 0 is opened For iL1 all j st wij gt 0 are connected to it So
vj = (service cost) + (fi +
zi)
j ljvj = fi + cij + i zi
Let L1 = i | yi = 1
Lj = i | xij = 1 L1 and lj = |Lj|
iL1 jiLj
j|iLj
jiLjiL1
Theory Seminar 042002
Finally hellip
Theorem Total cost le 4 times the optimal cost
Proof Total cost le
i fiyi + 3j (rj ndash lj)vj + fi + cij
facility cost of phase 1 cost for
copies connected by
phase 1
cost of phase 2
lei fiyi + 3j (rj ndash lj)vj + (j ljvj ndash i
zi )
lei fiyi + 3(j rjvj ndash i zi )
le4OPT
iL1 jiLj
Theory Seminar 042002
A Randomized Algorithm
Idea Open i with probability ρyi
Expected facility cost le ρi fiyi
Hope that each copy j(c) has a nearby facility open and service cost decreases
Not quitehellip no facility may be open
Cluster facilities open ge 1 facility in each cluster
Theory Seminar 042002
Phase 1 Pruning out ndashzis
Open all i st yi = 1
For each j if xij = yi = 1 connect j to i
Let Lj = i | xij = 1 and lj = |Lj|
Cost = j ljvj ndash i zi
Lj
Fj10
rrsquoj = residual reqmt = 6
Lrsquoj
Phase 2
Open all i st frac12 le yi lt 1
For each j let Lrsquoj = i | frac12 le xij lt 1
Connect copies of j to i Lrsquoj
Lose a factor of 2
facilities opened in
phases 1 2yi = 1 frac12 le yi lt 1yi lt frac12
Set L1
Set L2
Theory Seminar 042002
Phase 3
Notation facwt(S j) = iS xij
1 Form clusters Each cluster has facwt ge frac12
2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi
3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =
4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Finally hellip
Theorem Total cost le 4 times the optimal cost
Proof Total cost le
i fiyi + 3j (rj ndash lj)vj + fi + cij
facility cost of phase 1 cost for
copies connected by
phase 1
cost of phase 2
lei fiyi + 3j (rj ndash lj)vj + (j ljvj ndash i
zi )
lei fiyi + 3(j rjvj ndash i zi )
le4OPT
iL1 jiLj
Theory Seminar 042002
A Randomized Algorithm
Idea Open i with probability ρyi
Expected facility cost le ρi fiyi
Hope that each copy j(c) has a nearby facility open and service cost decreases
Not quitehellip no facility may be open
Cluster facilities open ge 1 facility in each cluster
Theory Seminar 042002
Phase 1 Pruning out ndashzis
Open all i st yi = 1
For each j if xij = yi = 1 connect j to i
Let Lj = i | xij = 1 and lj = |Lj|
Cost = j ljvj ndash i zi
Lj
Fj10
rrsquoj = residual reqmt = 6
Lrsquoj
Phase 2
Open all i st frac12 le yi lt 1
For each j let Lrsquoj = i | frac12 le xij lt 1
Connect copies of j to i Lrsquoj
Lose a factor of 2
facilities opened in
phases 1 2yi = 1 frac12 le yi lt 1yi lt frac12
Set L1
Set L2
Theory Seminar 042002
Phase 3
Notation facwt(S j) = iS xij
1 Form clusters Each cluster has facwt ge frac12
2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi
3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =
4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
A Randomized Algorithm
Idea Open i with probability ρyi
Expected facility cost le ρi fiyi
Hope that each copy j(c) has a nearby facility open and service cost decreases
Not quitehellip no facility may be open
Cluster facilities open ge 1 facility in each cluster
Theory Seminar 042002
Phase 1 Pruning out ndashzis
Open all i st yi = 1
For each j if xij = yi = 1 connect j to i
Let Lj = i | xij = 1 and lj = |Lj|
Cost = j ljvj ndash i zi
Lj
Fj10
rrsquoj = residual reqmt = 6
Lrsquoj
Phase 2
Open all i st frac12 le yi lt 1
For each j let Lrsquoj = i | frac12 le xij lt 1
Connect copies of j to i Lrsquoj
Lose a factor of 2
facilities opened in
phases 1 2yi = 1 frac12 le yi lt 1yi lt frac12
Set L1
Set L2
Theory Seminar 042002
Phase 3
Notation facwt(S j) = iS xij
1 Form clusters Each cluster has facwt ge frac12
2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi
3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =
4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Phase 1 Pruning out ndashzis
Open all i st yi = 1
For each j if xij = yi = 1 connect j to i
Let Lj = i | xij = 1 and lj = |Lj|
Cost = j ljvj ndash i zi
Lj
Fj10
rrsquoj = residual reqmt = 6
Lrsquoj
Phase 2
Open all i st frac12 le yi lt 1
For each j let Lrsquoj = i | frac12 le xij lt 1
Connect copies of j to i Lrsquoj
Lose a factor of 2
facilities opened in
phases 1 2yi = 1 frac12 le yi lt 1yi lt frac12
Set L1
Set L2
Theory Seminar 042002
Phase 3
Notation facwt(S j) = iS xij
1 Form clusters Each cluster has facwt ge frac12
2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi
3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =
4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Phase 3
Notation facwt(S j) = iS xij
1 Form clusters Each cluster has facwt ge frac12
2 Open facilities Open ge 1 facility in a cluster ndash used as a backup facility Open facility i with prob 2yi
3 Assign facilities to copies Each copy j(c) gets a preferred set of facilities ndash P(j(c)) with facwt ge frac12 For c d P(j(c)) P(j(d)) =
4 Connect clients Connect j(c) to the nearest i open in P(j(c)) or to a backup facility
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
ClusteringAfter phases 1 and 2 Fj = i | xij lt frac12 Sort these by cij and distribute among the rrsquoj copies
Cj(c) = avg service cost of the cth copy denote c Cj(c) = cijxij by Cj
Initial Fj before any iterations
Cj(1)
Cj(2)
Cj(3)3
i Fj
client j
Want the following properties
Clusters to be disjoint
Each cluster have facwt ge frac12
Each j be connected to rrsquoj clusters
iFj
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Iterative algorithm
S = j | rrsquoj gt 0
aj = lsquoactiversquo copy of j initially = 1
Ĉj(aj) = avg distance to the first k
facilities in Fj gathering facwt ge frac12
say these facilities lsquoserversquo j
Will maintain Ĉj(aj) le Cj(aj)
X
X
X1
Fj after some iterations
X i removed from Fj
i Fjserving jĈj(3)
facilities serving j
aj = 3
4X
(aj)
(aj)
(aj)
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Start of iteration
1 Choose j in S with minimum vj + Ĉj(aj)
2 Form cluster M = facilities serving j Note facilities are not split
3 For k st Fk M decrease rrsquok advance ak set Fk = Fk ndash M
2j(3) Cluster M
aj = 1
4
1
XX X
aj = 4 Cluster M
aj = 2
3
client in S
facility in some Fj
X facility removed from Fj
(aj)
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Opening Facilities
Central facilities opened in 2 steps
1 Open exactly 1 facility in M i opened with prob qyi Acts as backup
denoted b(k ) for each k st Fk M
2 Open each i in M indep with prob (2-q)yi and independent of step 1
Non-central facilities
Cluster M
k
open with prob 2yi independent of other choices
j
(ak
)
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Let Sj(c) = avg dist from j to P(j(c))
= ( cijxij)facwt(P(j(c))
j)
Then c Sj(c) le 2Cj
Distributing Facilities
iP(j(c))
j
rrsquoj = 3
P(j(1))P(j(2))
P(j(3))
Copy c gets a preferred set P(j(c))Preferred sets are disjoint
Ensure facwt(P(j(c)) j) ge frac12 for all c
Possible to do so since each xij lt frac12
facility in Fj
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Analysis
Feasibility follows from
1 Facilities in phases 1 2 not reused
2 After clustering j is connected to rrsquoj disjoint clusters backups are distinct
3 Preferred sets are disjoint
So j connected to rj distinct facilities
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Facility cost
Recall L1 = i | yi = 1
Phase 2 incur a factor of 2
Phase 3 each i is opened with probability 2yi
Expected facility cost le 2 fiyifor phases 2 3
iL1
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Bounding backup cost denoted by B rv
D event that no i in P(j(c)) is open
Lemma E[B|D] le 2vj + Cj(c)
Proof 2 cases
Service cost I
iM Fj st cik le Ĉj(d)
Also vk + Ĉj(d) le vj + Ĉj(c) le vj + Cj(c)
k(d)
j(c)
le vj
le Ĉj(d)le vk
B
k(d)
j(c)
le vj
le vk
iM Fj cik gt Ĉj(d)
le Ĉj(d) in expectatio
n
1)
2)
backup = b(j(c))
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Service Cost II
Fix j c Let X(c) = service cost of j(c)
Let di = cij pi = prob i is opened = 2yi
B(c) = backup costD(c) = event that no iP(j(c)) is
openp = Pr[D(c)] = (1-p1)hellip(1-pm) le e-1
davg = weighted avg of the dis
= (i pidi)(i pi) = Sj(c)
d1
d2 dm
P(j(c)) sorted by increasing cij
j(c)
i P(j(c))
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Then
E[X(c)] = [p1d1 + (1-p1)p2d2 + hellip
+ (1-p1)hellip(1-pm-1)pmdm]
+ pE[B(c)|D(c)]
le (1-p)davg + p[2vj + Cj(c)]
le (1-e-1)Sj(c) + e-1[2vj +
Cj(c)]
Let X = c X(c) = service cost of j
c Sj(c) le 2Cj and c Cj(c) le 2Cj
Summing over all c = 1helliprrsquoj
E[X] le (1-e-1)2Cj + e-1(2rrsquojvj + Cj)
le 2Cj + 2e-1rrsquojvj
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Putting it all together
Phase 1 pay the optimal LP cost
Phases 2 3
bull Facility cost twice LP facility cost
bull Service costLose a factor of 2 for phase 2Phase 3 cost is 2(LP service cost)+2e-1(dual value)
Overall cost for le (2+2e)(LP cost) phases 2 3
Total cost le (2+2e)OPT
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
How to improve this
bull Distribute facilities more equitably (in an expected sense) among copies - decreases prob of lsquobadrsquo event
bull Better analysis ndash maximum distance within a cluster can be bounded by 2Cj(c)
bull Balance phases 2 and 3
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Summary of Results
bullGive a 2076-approx algorithm for non-uniform rjs Based on LP rounding using complem slackness
bullFor rj = r extend the primal-dual algorithm of (JMS02) to get a 152-approximation
bullFault tolerant k medians with rj = r
a Primal-dual algorithm (JMS02) gives a 4-approx using Lagrangean relaxation
b LP rounding gives a factor of 8
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs
Theory Seminar 042002
Open Questions
1 Reduce gap between rj = r non-uniform rj
2 Combinatorial algorithms for non-uniform rj primal-dual local-search
3 Constant-factor approx for fault tolerant k medians with non-uniform rjs