Post on 05-Jan-2016
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New Algorithms for Disjoint Paths Problems
Sanjeev KhannaUniversity of Pennsylvania
Joint work with
Chandra Chekuri Bruce Shepherd
Edge Disjoint Path Problem (EDP)
Input: Graph G(V,E), node pairs s1t1, s2t2, ..., sktk
Goal: Route a maximum # of si-ti pairs using
edge-disjoint paths
s1
s2
s3
t1
t2
t3s4
t4
Edge Disjoint Path Problem (EDP)
Input: Graph G(V,E), node pairs s1t1, s2t2, ..., sktk
Goal: Route a maximum # of si-ti pairs using
edge-disjoint paths
s1
s2
s3
t1
t2
t3s4
t4
EDP on Stars
u1 u4
u2 u3
u1 u2 u3
u41 1
11
Matching in G EDP in Star
And vice versa.
Two Pair Problem
Input: Graph G(V,E) and two pairs s1t1, s2t2.
Goal: Can we simultaneously route s1 to t1 and s2 to t2
in an edge-disjoint manner?
Two Pair Problem
Input: Graph G(V,E) and two pairs s1t1, s2t2.
Goal: Can we simultaneously route s1 to t1 and s2 to t2
in an edge-disjoint manner?
NP-hard if G is a directed graph [Fortune, Hopcroft, Wylie’80].
Polynomial-time solvable for any constant number of pairs if G is undirected [Roberston, Seymour’88].
Routing Problems
Related problems: node disjoint paths each pair si-ti has a demand di and
edges/nodes have capacities
Fundamental to combinatorial optimization Applications to VLSI, network design and
routing, resource allocation & related areas Related to significant theoretical advances
Coping with Hardness
Settle for sub-optimal solutions: route only a fraction of the pairs that can be
routed in an optimal solution allow for small violations of edge capacities
Approximation algorithm A runs in polynomial time approximation ratio: how good is A approx ratio ) A(I) ¸ OPT(I)/ for all I Would like to be as small as possible
A Greedy Algorithm
Among the unrouted pairs, pick the pair that has the shortest path in the current graph.
Route this pair and remove all edges on the path from the graph.
Clearly gives an edge-disjoint routing.
How good is this algorithm?
Analysis of the Greedy Algorithm
n: # of vertices m: # of edges
Fix an optimal solution, say, OPT. If each greedy path is at most m1/2 edges long, it destroys at most m1/2 paths in OPT.
Suppose at some point, a path chosen by greedy is longer than m1/2. Since there are only m edges, OPT can chose at most m1/2 paths from here on.
So greedy gives an O(m1/2)-approximation.
A Bad Example
Greedy chooses the red path and none of the blue pairs
can be routed as a result.
Surely, we could do better ...
Not if the graph is directed!
[Guruswami, K, Rajaraman, Shepherd, Yannakakis ’99]
It is NP-hard to get an O(m1/2 - ) approximation for directed graphs for any > 0.
Surely, we could do better ...
Not if the graph is directed!
[Guruswami, K, Rajaraman, Shepherd, Yannakakis ’99]
It is NP-hard to get an O(m1/2 - ) approximation for directed graphs for any > 0.
[Chuzhoy, K ’05]For undirected graphs, O( log1/2- n)-approximation
is hard.(Builds on [Andrews, Zhang ’05] .)
Surely, we could do better ...
Not if the graph is directed!
[Guruswami, K, Rajaraman, Shepherd, Yannakakis ’99] It is NP-hard to get an O(m1/2 - ) approximation for directed graphs for any > 0.
[Chuzhoy, K ’05]For undirected graphs, O( log1/2- n)-approximation is
hard.(Builds on [Andrews, Zhang ’05] .)
The O(m1/2)-approximation is the best known in general (as a function of m).
All-or-Nothing Flow Prob (ANF)
Input: Graph G(V,E), node pairs s1t1, s2t2, ..., sktk.
Goal: Route a maximum # of si-ti pairs such that each routed pair has one unit of flow.
s1 s2
t1t2
s1
s2
t1t2
1/2
1/2
1/2
1/2
Recent Progress
O(log2 n) approximation for ANF in undirected graphs.
[Chuzhoy, K ’05] O( log1/2- n)-approximation is hard.
[Chekuri, K, Shepherd: ’04 and ’05]
Recent Progress
O(log2 n) approximation for ANF in undirected graphs.
[Chuzhoy, K ’05] O( log1/2- n)-approximation is hard.
O(log n) approximation for EDP in planar undirected graphs when up to two paths can share an edge.
[Chekuri, K, Shepherd: ’04 and ’05]
Recent Progress
O(log2 n) approximation for ANF in undirected graphs.
[Chuzhoy, K ’05] O( log1/2- n)-approximation is hard.
O(log n) approximation for EDP in planar undirected graphs when up to two paths can share an edge.
Similar results for the node-disjoint versions as well as versions with arbitrary demands and capacities.
[Chekuri, K, Shepherd: ’04 and ’05]
Recent Progress
O(log2 n) approximation for ANF in undirected graphs.
[Chuzhoy, K ’05] O( log1/2- n)-approximation is hard.
O(log n) approximation for EDP in planar undirected graphs when up to two paths can share an edge.
Similar results for the node-disjoint versions as well as versions with arbitrary demands and capacities.
Previous algorithms had (n1/2) approximation ratio.
[Chekuri, K, Shepherd: ’04 and ’05]
Rest of the Talk
EDP in planar graphs A fractional relaxation A new framework for routing problems Well-linked sets and crossbars Routing using crossbar structures
EDP with congestion in general graphs
Multicommodity Flow Formulation (LP)
Routing is relaxed to be a flow from si to ti. A pair can be routed for a fractional amount.
xi : fraction of si-ti flow that is routed.
Max i xi s.t. 8 e total flow through e · 1.
0 · xi · 1.
Randomized Rounding
[Raghavan and Thompson ’87] For each pair (si,ti), decompose the flow xi 2 [0,1]
into a collection of flow paths. Decide to route pair (si,ti) with probability xi. If yes, choose an si-ti flow path with probability
proportional to the flow on it.
O(n1/c)-approximation if we allow up to c paths to use
an edge.
How Good is this LP?
[GVY ’93]
s1s2si s3sk-1sk
t1
tk-1
tk
t3
t2
ti
How Good is this LP?
[GVY ’93]
s1s2si s3sk-1sk
t1
tk-1
tk
t3
t2
ti
Gap holds for planar graphs
n1/2) Lower Bound
How Good is this LP?
[GVY ’93]
s1s2si s3sk-1sk
t1
tk-1
tk
t3
t2
ti
Gap holds for planar graphs
n1/2) Lower Bound
O(n2/3) Upper Bound
A New Framework
1. Solve the LP relaxation.
2. Use LP solution to decompose input instance into a collection of instances with special structure, called well-linked instances.
3. Well-linked instances have special properties; use them for routing!
High-level Algorithm for Planar Graphs
1. Solve the LP relaxation.
2. Use LP solution to decompose input instance into a collection of well-linked instances.
3. Well-linked planar instances have crossbars, use them for routing!
Assume w.l.o.g. input graph to be bounded degree.
Well-linked Set
Subset X is well-linked in G if for every partition (S,V-S) ,
# of edges cut is at least # of X vertices in smaller side.
S V - S
For all S ½ V with |S Å X| · |X|/2, |(S)| ¸ |S Å X|
Instance of EDP
Input instance: G, X, M
G : underlying graph.X : {s1, t1, s2, t2, ..., sk, tk} is the terminal set
M : a matching on X , namely, (s1,t1), (s2,t2) ... (sk,tk)
that needs to be routed in G.
Well-linked Instance of EDP
Input instance: G, X, M
G : underlying graph.X : {s1, t1, s2, t2, ..., sk, tk} is the terminal set
M : a matching on X , namely, (s1,t1), (s2,t2) ... (sk,tk) that needs to be routed in G.
X is well-linked in G.
Examples
s1 t1
s2 t2
s3 t3
s4 t4
Not a well-linked instance
s1 t1
s2 t2
s3 t3
s4 t4
A well-linked instance
H=(V,E) is a cross-bar with respect to an interface I µ V
if any matching on I can be routed using edge-disjoint
paths.
Ex: a complete graph is a cross-bar with I=V
Crossbars
H
Grids as Crossbars
s1 s3s2s4 s5t1 t2 t3 t4 t5
First row is interface
Grids in Planar Graphs
Theorem [Robertson, Seymour, Thomas ’94]: If G is a planar graph with a well-linked set of size k, then G has a grid minor H of size (k) as a subgraph.
v Gv
Grid minor is a crossbar with congestion 2[Kleinberg ’96]: uses it for half disjoint paths.
Gv
Routing pairs in X using H
X
H
Routing pairs in X using H
X
H
A Single-Source Single-Sink Flow Computation
Sink
Source
Routing pairs in X using H
X
H
Route X to I
Sink
Source
Routing pairs in X using H
X
H
Route X to I
Routing pairs in X using H
X
H
Route X to I and use H for pairing up
Several Technical Issues
H is smaller than X, so can pairs reach H?
What if X cannot reach H?
Can X reach interface of H without using edges of H?
Can H be found in polynomial time?
Routing Pairs to I
Claim: If some subset A of terminals can reach I,
then any subset A’ of terminals with |A’| · |A|/2
can reach I.
Use the fact that the terminals are well-linked.
Routing Pairs to I
A’ I
V - S
S
p edges
Routing Pairs to I
A’ I
V - S
S
p edges
If |S Å A| ¸ |A|/2, then p ¸ |A|/2 ¸ |A’| since A can reach B.
A A
Routing Pairs to I
A’ I
V - S
S
p edges
If |(V-S) Å A| ¸ |A|/2, then p ¸ |A|/2 since terminals are well-linked.
AA
Routing Pairs to I
A’ I
V - S
S
p ¸ |A’| edges
Thus A’ can be routed to I.
We can choose any |A’|/2 pairs to be routed to I.
Summarizing ...
1. Solve the LP relaxation.
2. Use LP solution to decompose input instance into a collection of well-linked instances.
3. Well-linked instances on planar graphs have a grid crossbar. Use it to route many pairs.
Can Route the Entire Matching
For EDP, suffices to route simply a constant fraction of pairs in the EDP instance.
Actually, we can route the entire matching with O(1) congestion.
Decomposition into Well-linked Instances
G
G1 G2 Gr
Xi is well-linked in Gi
i |Xi| ¸ OPT/
Example
s1 t1
s2 t2
s3 t3
s4 t4
s1 t1
s2 t2
s3 t3
s4 t4
Decomposition
= O(log2 n) in general graphs. = O(log n) for planar graphs.
Decomposition based on LP solution.
EDP with Congestion in General Graphs
What is the integrality gap of multicommodity flow relaxation when 2 paths can share an edge?
Is it more than a constant?
How Good is this LP?
s1s2si s3sk-1sk
t1
tk-1
tk
t3
t2
ti
Gap holds for planar graphs
EDP with Congestion in General Graphs
What is the integrality gap of multicommodity flow relaxation when 2 paths can share an edge?
Is it more than a constant?
[Chuzhoy, K ’05]A simple family of instances with gap roughly log1/c n for anycongestion c.
Shows that there is a superconstant gap between fractional flow
and 1/c-integral flow for any integer c.
Similar gap results hold even for All-or-Nothing Flow.
EDP with Congestion in General Graphs
EDP in general undirected graphs is log(1/c) n-hard to approximate when congestion of c is allowed.
[Andrews, Zhang’05], [Chuzhoy, K ’05], [Guruswami, Talwar’05].
EDP with Congestion in General Graphs
EDP in general undirected graphs is log(1/c) n-hard to approximate when congestion of c is allowed.
[Andrews, Zhang’05], [Chuzhoy, K ’05], [Guruswami, Talwar’05].
On the positive side, when congestion of c is allowed O(n1/c)-approximation is known [Srinivasan’ 97], [Baveja-Srinivasan’ 00], [Azar-Regev’ 01].
Summary
New algorithmic framework for routing problems.
Recent developments for undirected graphs O(log2 n)-approximation for even degree planar graphs with
no congestion [Kleinberg ’05] O(1)-approximation for planar graphs with O(1)-congestion
[Chekuri, K, Shepherd ’05] Tight bound of (n1/2) for LP integrality gap in general graphs,
improves upon the previous upper bound of O(n^{2/3}). [Chekuri, K,
Shepherd ’05]
Many open problems …
Thank You!
Routing Using the Grid Crossbar
Let k = |X| be the # of terminals.
Suppose we have a k/10 by k/10 grid crossbar H.
Route terminals in X to interface vertices in H: a single source-single sink max flow computation.
If max flow ¸ k/100, we can route in an edge-disjoint manner at least k/100 terminals to the interface and pair them using the grid.
EDP on Line Networks
si ti
Independent set in interval graphs.
Advantage of Well-linkedness
LP value does not depend on input pairing M.
s1 t1
s2 t2
s3 t3
s4 t4
Claim: If X is well-linked, then for any pairing on X, LP value is (|X|/log |X|). We have symmetry w.r.t. to every pairing.