On “real” and on “bad” LPs
. . . but when they had taken their seats Her Majestyturned to the president and resumed.
“. . . was he nevertheless as bad as he was painted?Or more to the point,” and she took up her soup spoon,
“was he as good?”
Alan Bennett: The Uncommon Reader, 2007
On “real” and on “bad” LPs
. . . but when they had taken their seats Her Majestyturned to the president and resumed.
“. . . was he nevertheless as bad as he was painted?Or more to the point,” and she took up her soup spoon,“was he as good?”
Alan Bennett: The Uncommon Reader, 2007
On “real” LPs
afiro.lp:32 variables8 equations
dimension: 2419 explicit inequality constraints32 non-negativities29 facets
On “real” LPs
afiro.lp:32 variables8 equationsdimension: 24
19 explicit inequality constraints32 non-negativities29 facets
On “real” LPs
afiro.lp:32 variables8 equationsdimension: 2419 explicit inequality constraints32 non-negativities
29 facets
On “real” LPs
afiro.lp:32 variables8 equationsdimension: 2419 explicit inequality constraints32 non-negativities29 facets
On “real” LPs\Problem name: afiro.lp
MinimizeCOST: - 0.4 X02 - 0.32 X14 - 0.6 X23 - 0.48 X36 + 10 X39
Subject ToR09: - X01 + X02 + X03 = 0R10: - 1.06 X01 + X04 = 0X05: X01 <= 80X21: - X02 + 1.4 X14 <= 0R12: - X06 - X07 - X08 - X09 + X14 + X15 = 0R13: - 1.06 X06 - 1.06 X07 - 0.96 X08 - 0.86 X09 + X16 = 0X17: X06 - X10 <= 80X18: X07 - X11 <= 0X19: X08 - X12 <= 0X20: X09 - X13 <= 0R19: - X22 + X23 + X24 + X25 = 0R20: - 0.43 X22 + X26 = 0X27: X22 <= 500X44: - X23 + 1.4 X36 <= 0R22: - 0.43 X28 - 0.43 X29 - 0.39 X30 - 0.37 X31 + X38 = 0R23: X28 + X29 + X30 + X31 - X36 + X37 + X39 = 44X40: X28 - X32 <= 500X41: X29 - X33 <= 0X42: X30 - X34 <= 0X43: X31 - X35 <= 0X45: 2.364 X10 + 2.386 X11 + 2.408 X12 + 2.429 X13 - X25 + 2.191 X32
+ 2.219 X33 + 2.249 X34 + 2.279 X35 <= 0X46: - X03 + 0.109 X22 <= 0X47: - X15 + 0.109 X28 + 0.108 X29 + 0.108 X30 + 0.107 X31 <= 0X48: 0.301 X01 - X24 <= 0X49: 0.301 X06 + 0.313 X07 + 0.313 X08 + 0.326 X09 - X37 <= 0X50: X04 + X26 <= 310X51: X16 + X38 <= 300
End
On “real” LPs: afiro.lp
afiro.lp:dimension: 2429 facets
1654 vertices78 degenerate verticesminimal vertex degree = 24maximal vertex degree = 39average vertex degree = 24.7120433 edges11718 horizontal edges (more than half of the edges horizontal!)maximal vertex: unique, degree 39minimal vertex: 4 of them, 2-facegraph distance minimal vertices to maximal vertex: 2graph diameter: 5 (Hirsch conjecture sharp!)
On “real” LPs: afiro.lp
afiro.lp:dimension: 2429 facets
1654 vertices78 degenerate verticesminimal vertex degree = 24maximal vertex degree = 39average vertex degree = 24.71
20433 edges11718 horizontal edges (more than half of the edges horizontal!)maximal vertex: unique, degree 39minimal vertex: 4 of them, 2-facegraph distance minimal vertices to maximal vertex: 2graph diameter: 5 (Hirsch conjecture sharp!)
On “real” LPs: afiro.lp
afiro.lp:dimension: 2429 facets
1654 vertices78 degenerate verticesminimal vertex degree = 24maximal vertex degree = 39average vertex degree = 24.7120433 edges11718 horizontal edges (more than half of the edges horizontal!)
maximal vertex: unique, degree 39minimal vertex: 4 of them, 2-facegraph distance minimal vertices to maximal vertex: 2graph diameter: 5 (Hirsch conjecture sharp!)
On “real” LPs: afiro.lp
afiro.lp:dimension: 2429 facets
1654 vertices78 degenerate verticesminimal vertex degree = 24maximal vertex degree = 39average vertex degree = 24.7120433 edges11718 horizontal edges (more than half of the edges horizontal!)maximal vertex: unique, degree 39minimal vertex: 4 of them, 2-face
graph distance minimal vertices to maximal vertex: 2graph diameter: 5 (Hirsch conjecture sharp!)
On “real” LPs: afiro.lp
afiro.lp:dimension: 2429 facets
1654 vertices78 degenerate verticesminimal vertex degree = 24maximal vertex degree = 39average vertex degree = 24.7120433 edges11718 horizontal edges (more than half of the edges horizontal!)maximal vertex: unique, degree 39minimal vertex: 4 of them, 2-facegraph distance minimal vertices to maximal vertex: 2graph diameter: 5 (Hirsch conjecture sharp!)
On “real” LPs: afiro.lp
-1200
-1000
-800
-600
-400
-200
0
200
-500 0 500 1000 1500 2000 2500 3000 3500
afiro
On “real” LPs: afiro.lp
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
random1random2
On “real” LPs: boeing1.lp
0
10000
20000
30000
40000
50000
60000
70000
-300 -200 -100 0 100 200 300
boeing1
On “real” LPs: boeing2.lp
-82000
-80000
-78000
-76000
-74000
-72000
-70000
-68000
-300 -250 -200 -150 -100
boeing2
On “real” LPs: fit1d.lp
-20000
-10000
0
10000
20000
30000
40000
50000
60000
-10000 0 10000 20000 30000 40000 50000 60000 70000 80000
fit1d
On “real” LPs: kb2.lp
-14000
-12000
-10000
-8000
-6000
-4000
-2000
0
-1800 -1600 -1400 -1200 -1000 -800 -600 -400 -200 0
kb2
On “real” LPs: sc105.lp
-5000
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
-50 -40 -30 -20 -10 0
sc105
On “bad” 3D LPs. . . joint work with
Volker Kaibel, Rafael Mechteland Micha Sharir
(SIAM J. Comp. 2005)
On “bad” 3D LPs
Geometry vs. Combinatorial Model [Steinitz 1922]:
3-polytope P planar, 3-connected graph G(no loops, no parallel edges)
On “bad” 3D LPs
Geometry vs. Combinatorial Model [Mihalisin & Klee 2000]:
vmin
vmax
ϕ
3-polytope Pgeneric linear functionϕ : R3 → R
vminvmax
• 3-polytopal graph G
• acyclic orientation withunique sink in every face (AOF)
• three disjoint monotone pathsfrom vmax to vmin
On “bad” 3D LPs
Geometry vs. Combinatorial Model [Mihalisin & Klee 2000]:
vmin
vmax
ϕ
3-polytope Pgeneric linear functionϕ : R3 → R
vminvmax
• 3-polytopal graph G
• acyclic orientation withunique sink in every face (AOF)
• three disjoint monotone pathsfrom vmax to vmin
On “bad” 3D LPs: RandomEdgeRandomEdge takes a step to an improving neighbor chosenuniformly at random:
vminvstart
vmax
Theorem:The worst-case running time of RandomEdge on a 3D polytopewith n facets is bounded by
1.3473 n ≤ RandomEdge(n) ≤ 1.4943 n.
On “bad” 3D LPs: RandomEdgeRandomEdge takes a step to an improving neighbor chosenuniformly at random:
vminvstart
vmax
Theorem:The worst-case running time of RandomEdge on a 3D polytopewith n facets is bounded by
1.3473 n ≤ RandomEdge(n) ≤ 1.4943 n.
On “bad” 3D LPs: RandomEdge
For n ≤ 12, we enumerated 3-connected cubic graphs with nfaces using plantri by [Brinkmann & McKay]
. . . and all the “abstract objective functions” on each ofthese . . .
. . . to see what worst-case examples look like:
vminvstartvmax
On “bad” 3D LPs: RandomEdgethe “backbone”:
v0 = vmin
v1v2vk−1 vk−2 vk−3vmax
a configuration:
vi ,min
vi ,start
4
4
622
63
3
58
flow costs per configuration:438
facets per configuration: 3 + 1
=⇒ RandomEdge(n)/n ≥ 438 · 4
≈ 1.3437
On “bad” 3D LPs: RandomEdgethe “backbone”:
v0 = vmin
v1v2vk−1 vk−2 vk−3vmax
a configuration:
vi ,min
vi ,start
4
4
622
63
3
58
flow costs per configuration:438
facets per configuration: 3 + 1
=⇒ RandomEdge(n)/n ≥ 438 · 4
≈ 1.3437
On “bad” 3D LPs: RandomEdgethe “backbone”:
v0 = vmin
v1v2vk−1 vk−2 vk−3vmax
a configuration:
vi ,min
vi ,start
4
4
622
63
3
58
flow costs per configuration:438
facets per configuration: 3 + 1
=⇒ RandomEdge(n)/n ≥ 438 · 4
≈ 1.3437
On “bad” 3D LPs: RandomEdge
a worse configuration:
flow costs per configuration:1897128
facets per configuration: 10 + 1
=⇒ RandomEdge(n)/n ≥ 18971408
≈ 1.3473.
On “bad” 3D LPs: RandomEdge
Recursion formula for the expected number of pivot steps E(v)“starting from v ”:
E(v) = 1 +1
| δ+(v)|∑
u:(v ,u)∈δ+(v)
E(u),
vmin
vstartvmax
0132
52
72
134
358
On “bad” 3D LPs: RandomEdge
Recursion formula for the expected number of pivot steps E(v)“starting from v ”:
E(v) = 1 +1
| δ+(v)|∑
u:(v ,u)∈δ+(v)
E(u),
vmin
vstartvmax
0
132
52
72
134
358
On “bad” 3D LPs: RandomEdge
Recursion formula for the expected number of pivot steps E(v)“starting from v ”:
E(v) = 1 +1
| δ+(v)|∑
u:(v ,u)∈δ+(v)
E(u),
vmin
vstartvmax
01
32
52
72
134
358
On “bad” 3D LPs: RandomEdge
Recursion formula for the expected number of pivot steps E(v)“starting from v ”:
E(v) = 1 +1
| δ+(v)|∑
u:(v ,u)∈δ+(v)
E(u),
vmin
vstartvmax
0132
52
72
134
358
On “bad” 3D LPs: RandomEdge
Recursion formula for the expected number of pivot steps E(v)“starting from v ”:
E(v) = 1 +1
| δ+(v)|∑
u:(v ,u)∈δ+(v)
E(u),
vmin
vstartvmax
0132
52
72
134
358
On “bad” 3D LPs: RandomEdge
Recursion formula for the expected number of pivot steps E(v)“starting from v ”:
E(v) = 1 +1
| δ+(v)|∑
u:(v ,u)∈δ+(v)
E(u),
vmin
vstartvmax
0132
52
72
134
358
On “bad” 3D LPs: RandomEdge
Recursion formula for the expected number of pivot steps E(v)“starting from v ”:
E(v) = 1 +1
| δ+(v)|∑
u:(v ,u)∈δ+(v)
E(u),
vmin
vstartvmax
0132
52
72
134
358
On “bad” 3D LPs: RandomEdge
Recursion formula for the expected number of pivot steps E(v)“starting from v ”:
E(v) = 1 +1
| δ+(v)|∑
u:(v ,u)∈δ+(v)
E(u),
vmin
vstartvmax
0132
52
72
134
358
On “bad” 3D LPs: RandomEdge
Theorem:The worst-case running time of RandomEdge on a 3D polytopewith n facets is bounded by
1.3473 n ≤ RandomEdge(n) ≤ 1.4943 n.
Proof of Upper bound:Complicated, LP-optimized inductionbased on the numbers of 1- and 2-vertices “ahead”by case analysis of possible local structure.
RandomEdge on KM-cubes
[Klee & Minty 1972]:
min xn : 0 ≤ x1 ≤ 1,εxi−1 ≤ xi ≤ 1− εxi−1 for 2 ≤ i ≤ n.
RandomEdge on KM-cubes
[Klee & Minty 1972] – Geometry:
“deformed product” [Amenta & Z. 1998] [Sanyal & Z. 2010]
. . . not a projective cube, cf. [Gritzmann & Klee 1993]
RandomEdge on KM-cubes
[Klee & Minty 1972] – Geometry:
“deformed product” [Amenta & Z. 1998] [Sanyal & Z. 2010]
. . . not a projective cube, cf. [Gritzmann & Klee 1993]
RandomEdge on KM-cubes[Klee & Minty 1972] – Combinatorial model:
11010100010101010111001↓
11010100001010101000110↓
11010100001010010111001↓
10101011110101101000110↓
10101011001010010111001↓
10101010110101101000110↓
. . .
RandomEdge on KM-cubes[Klee & Minty 1972] – Combinatorial model:
11010100010101010111001↓
11010100001010101000110↓
11010100001010010111001↓
10101011110101101000110↓
10101011001010010111001↓
10101010110101101000110↓
. . .
RandomEdge on KM-cubes[Klee & Minty 1972] – Combinatorial model:
11010100010101010111001↓
11010100001010101000110↓
11010100001010010111001↓
10101011110101101000110↓
10101011001010010111001↓
10101010110101101000110↓
. . .
RandomEdge on KM-cubes[Klee & Minty 1972] – Combinatorial model:
11010100010101010111001↓
11010100001010101000110↓
11010100001010010111001↓
10101011110101101000110↓
10101011001010010111001↓
10101010110101101000110↓
. . .
RandomEdge on KM-cubes[Klee & Minty 1972] – Combinatorial model:
11010100010101010111001↓
11010100001010101000110↓
11010100001010010111001↓
10101011110101101000110↓
10101011001010010111001↓
10101010110101101000110↓
. . .
RandomEdge on KM-cubes[Klee & Minty 1972] – Combinatorial model:
11010100010101010111001↓
11010100001010101000110↓
11010100001010010111001↓
10101011110101101000110↓
10101011001010010111001↓
10101010110101101000110↓
. . .
RandomEdge on KM-cubes
Recursion for running time of RandomEdge:
E(v) = 1 +1
| δ+(v)|∑
u:(v ,u)∈δ+(v)
E(u),
Conjectures/claims to deal with:
E(KMn) = O(n log2 n) [Kelly 1981]
E(KMn) = O(n) [Papadimitriou & Steiglitz 1982]
Worst starting vertex is “111111111” [Kelly 1981]
RandomEdge on KM-cubes
Recursion for running time of RandomEdge:
E(v) = 1 +1
| δ+(v)|∑
u:(v ,u)∈δ+(v)
E(u),
Conjectures/claims to deal with:
E(KMn) = O(n log2 n) [Kelly 1981]
E(KMn) = O(n) [Papadimitriou & Steiglitz 1982]
Worst starting vertex is “111111111” [Kelly 1981]
RandomEdge on KM-cubes
Theorem: [Gärtner, Henk & Z. 1998]
n2
4 ln n≤ E(KMn) ≤ .32 n2
Theorem: [Balogh & Pemantle 2007]
E(KMn) = Θ(n2)
RandomEdge on KM-cubes
Theorem: [Gärtner, Henk & Z. 1998]
n2
4 ln n≤ E(KMn) ≤ .32 n2
Theorem: [Balogh & Pemantle 2007]
E(KMn) = Θ(n2)
RandomEdge on KM-cubesTheorem: [Gärtner, Henk & Z. 1998]For n = 18, the “worst” starting vertex is not “111111111”,13616631777775844683363020123127094102813560533249201592684030464015036597749164177937892497054636200148618292482860171750895424612282188029893901861623777485911206277962056237212747724106971466780610966750343428884227862863263992240626944189474092321690578465850140349559978346803788901676121822279514510337366767357850445503507313736074535300137075905658614012808891118656316408374481670049167271660724973549966381898781024671174519123371033460330957586555415033425077727004460772400592700649563057465490900379364876075017619331015784049036558215285038489260535168237679310834438980211261828416142728106103764871709282502595038844295608788896317984140105479983862039598375987130214282003033020521051527717363707514678161308836221258641232146849278090766755354459331000620203946310574593680591647128707656257843986034927310159488243185598011054353166122873565317236418489392762400450466591101169432128401829430964496067737309072179851815110185854606923338019104432978507203552547721528956985400094903678817126979242634733
/24962914469583191566367899923960035966404195848061867674084257325351443541635707108155961226253248787890779064193993572350407540882488313462480442009188322965308175793834908350711775470723063600215807657623374362126466957217083277804322543021220684978006202697645006112925727562688499951019008352273202998698433082923670410460471087620616165741606476723494106039072694978995397627362457886096950309963504449900266130281466812904611199098539805421932601086188714264657278379296255216638772840224103572864219332581506845376087212926568647114428532255047582757200833981470518008818805688306634147913338380858098047897556191777014597833213397490582583305285642961484103228645506879443810132259225138635152335311879233430440856986173430296778743350033604723187899699712826261542556246067548824288988199191266967980956809764206176935319256985166746436097191235722924204644128666213286229783071506832436219870537040937793596789158704489369975002974080695684303196676875492248579163093862959138359780655177475508923698668685694367
*** ROUNDBF turned on to increase accuracy
54.5474439468
2#101010101010101010
RandomEdge on KM-cubesTheorem: [Gärtner, Henk & Z. 1998]For n = 18, the “worst” starting vertex is not “111111111”, but
13616684425655993868809930342829681416513745972117542059691282558789265359381405592027738235067168526490944745020032821831458348414708370359959721284479519471305069796673823323749468285051705771273197431466120774988286761021877147886424485186429677466830230656505140112567299734199737747316914047149456525668905990843395292570627973505504153803354783689366237724788621187438316245307594694578844829661290767878882426158862400116464580283005287051721523550761838202480537960286313206215370104486533179185155951187425478644148272361015209552862194206039184472942296180543643639536783031340491471882644253089850167358829961270644689650011819021006889213587618695068079416895590094361423303966234997939240089858719693593594374858225574919421676770475805291256813831033322745894364021986396433890467502668449390681018004891572703264811384653350462845288862763661943911510871076870345393271182732960163244935717489369649608477918928915351385767044907310095700518582096630823172413069321398119400382068757071877644544929382637646
/24962914469583191566367899923960035966404195848061867674084257325351443541635707108155961226253248787890779064193993572350407540882488313462480442009188322965308175793834908350711775470723063600215807657623374362126466957217083277804322543021220684978006202697645006112925727562688499951019008352273202998698433082923670410460471087620616165741606476723494106039072694978995397627362457886096950309963504449900266130281466812904611199098539805421932601086188714264657278379296255216638772840224103572864219332581506845376087212926568647114428532255047582757200833981470518008818805688306634147913338380858098047897556191777014597833213397490582583305285642961484103228645506879443810132259225138635152335311879233430440856986173430296778743350033604723187899699712826261542556246067548824288988199191266967980956809764206176935319256985166746436097191235722924204644128666213286229783071506832436219870537040937793596789158704489369975002974080695684303196676875492248579163093862959138359780655177475508923698668685694367
54.5476548512
2#101010101010101011
RandomEdge on KM-cubes
Conclusions:
RandomEdge is charming and simple, but awful to analyze.
We don’t understand really bad linear programs, e.g. indimension 4.
Experiments help.
RandomEdge on KM-cubes
Conclusions:
RandomEdge is charming and simple, but awful to analyze.
We don’t understand really bad linear programs, e.g. indimension 4.
Experiments help.
RandomEdge on KM-cubes
Conclusions:
RandomEdge is charming and simple, but awful to analyze.
We don’t understand really bad linear programs, e.g. indimension 4.
Experiments help.
Special Issue:Xxxxx Xxxxx Xxxxxx XxxxxxXxxxxxx
454 • ISSN 0179-537645(1) 001-000 (2011)
Springer
An International Journal of Mathematics and Computer Science
Founding EditorsJacob E. GoodmanRichard Pollack
Edited byHerbert Edelsbrunner
János PachGünter M. Ziegler
Volume 45 • Number 1 January 2011
DCGDiscrete & Computational Geomet ry
Discrete &
Computational Geom
etry
VOLUME
45
—
NUMBER
1
—
JANUARY
2011