New Algorithms for Disjoint Paths Problems

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.



[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] .)



[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(log n) approximation for EDP in planar undirected graphs when up to two paths can share an edge.


Recent Progress




Similar results for the node-disjoint versions as well as versions with arbitrary demands and capacities.


Recent Progress




Similar results for the node-disjoint versions as well as versions with arbitrary demands and capacities.

Previous algorithms had (n1/2) approximation ratio.


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


[GVY ’93]

s1s2si s3sk-1sk

t1

tk-1

tk

t3

t2

ti

Gap holds for planar graphs

n1/2) Lower Bound


[GVY ’93]

s1s2si s3sk-1sk

t1

tk-1

tk

t3

t2

ti


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


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


X

H

A Single-Source Single-Sink Flow Computation

Sink

Source


X

H

Route X to I

Sink

Source


X

H

Route X to I


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 ...


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?


s1s2si s3sk-1sk

t1

tk-1

tk

t3

t2

ti



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 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 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.

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New Algorithms for Disjoint Paths Problems