All Rights Reserved © Alcatel-Lucent 2006, ##### Matthew Andrews, Alcatel-Lucent Bell Labs Princeton Approximation Workshop June 15, 2011 Edge-Disjoint.

All Rights Reserved © Alcatel-Lucent 2006, #####

Matthew Andrews, Alcatel-Lucent Bell Labs

Princeton Approximation Workshop June 15, 2011

Edge-Disjoint Paths with Congestion

Edge Disjoint Paths (EDP)

• This talk– Routing on Edge-Disjoint Paths– Useful primitive for many industrial routing problems


• Input – Graph G (M edges, N nodes);– A set of demands, (si , ti);

• Output– A subset of demands routed on edge-disjoint paths;– Maximize such a subset

s

t

t

s

OPT = 1



• Output– A subset of demands routed on edge-disjoint paths;– Maximize such a subset

s

t

t

s

OPT = 1

NP-hard problemOne of the first!! (Karp’s

list)

http://datamining.typepad.com/photos/uncategorized/ljviz.png

Congestion Minimization


• Output– Route all demands;– Minimize max number of demand routes per edge.

s

t

t

s

OPT = 2

Edge Disjoint Paths with Congestion (EDPwC)

• Input – Graph G (M edges, N nodes);– A set of demands, (si , ti);– Congestion parameter c;

• Output– A subset of demands routed such that max congestion c

– If ALG can route X/ demands with congestion c whenever OPT can route X demands with congestion 1 then

ALG is an -approx with congestion c

s

t

t

s

Known Results

• Undirected Graphs – EDP solvable in polytime if the number of demands is constant– Robertson-Seymour

• Directed Graphs– NP-hard even for 2 demands– Fortune-Hopcroft-Wyllie

Known Results (Undirected Graphs)• Positive

– N1/2 - approx with congestion 1Chekuri-Khanna-Shepherd

– 1-approx with congestion log nRaghavan-Thompson

– N1/c - approx with congestion cAzar-Regev, Baveja-Srinivasan, Kolliopoulos-Stein

– polylog(N) - approx with congestion poly(log log N)

A

• Negative– No log(1-)/(c+1) N - approx with congestion c– A-Chuzhoy-Guruswami-Khanna-Talwar-Zhang






A

• Negative– No log0.5- N - approx with congestion 1– A-Chuzhoy-Guruswami-Khanna-Talwar-Zhang






A

• Negative– No 1 - approx with congestion (log log N)1-

– A-Chuzhoy-Guruswami-Khanna-Talwar-Zhang






A







A


Open Question:Is there a polylog(N)-approx with constant congestion?





– log61(N)-approx with (log log N)6 congestion A










This talk

Other Known Results

• Planar Graphs – O(1) - approx with congestion 4 – Chekuri-Khanna-Shepherd

• All-or-Nothing Flow (fractional paths allowed)– polylog(N) - approx with congestion 1– Chekuri-Khanna-Shepherd

Known Results (Directed Graphs)

• Positive – N 1/c - approx with congestion c – Azar-Regev, Baveja-Srinivasan, Kolliopoulos-Stein

• Negative– No N Ω(1/c) - approx with congestion c– A-Zhang, Chuzhoy-Guruswami-Khanna-Talwar

EDPwC in Graphs with Short Paths

• Fractional routing – If we allow fractional paths, problem is

a linear program

• Known result #1– If fractional path length is polylog(N), can route with congestion

loglog N

– Why?– Use randomized rounding. Each edge is dependent on polylog(N)

other edges

– Apply Lovasz Local Lemma

s t

ts

1/2

1/2

1/2

1/2

Graph Expansion

• Out-degree– Let out(S) = set of edges with one endpoint in S

– Abuse: out(S) = | out(S) |

• Expander– Graph G is an expander if

out(S) / |S|

is large whenever S is small

EDPwC in Expanders

• Known result #2– EDP has polylog(N)-approx in expanders

– e.g. Broder-Frieze-Upfal, Kolman-Scheideler

– Even better, can connect polylog(N) fraction of ANY set of terminals using disjoint paths

• Why?– Using random walks, can connect any

pair of terminals with paths of polylog(N) length

– Produces good fractional solution

– Round fractional solution via Lovasz Local Lemma

Rao-Zhou

• Known result #3– (Rao-Zhou) polylog(N) –approx on disjoint paths if min-cut in

graph has size polylog(N)








Hardness of EDPwC

• Hardness Idea 1– If routing is fixed then there is reduction from Max Independent

Set

Hardness of EDPwC

• Hardness Idea 2– Embed hardness construction into expander (high-girth!)

– Terminals are close together (distance log1/2 N )

– Each demand has only 1 short ( canonical ) path

– Low congestion soln must use canonical paths

– Routing component is removed– Use previous reduction from

Max-Indpt-Set

– Use product of regular expander (e.g. random graph) with Max-Indpt-Set example








Rao-Zhou

• Known result #3– (Rao-Zhou) polylog(N) –approx on disjoint paths if min-cut in

graph has size polylog(N)

Rao-Zhou Analysis

• Why?– (Khandekar-Rao-Vazirani) Can build expander on | T | terminals

using polylog( | T | ) bipartite matchings

Rao-Zhou Analysis

• Application to EDPwC– Use max-flows to find matchings between terminals

– How to bound congestion from max-flows ?

(Max flows exist due to linkedness results of Chekuri-Khanna-Shepherd)

• Partitioning– Randomly partition edges into polylog( | T | ) pieces– Solve max-flow in each piece

– How do we know this is feasible?

• p-skeletons– (Karger sparsification) If min-cut in graph is large, all cuts are

preserved in each component up to polylog( | T | ) factors

– Can connect “enough” terminals using max-flow-min-cut thm

Rao-Zhou Analysis

• Build expanders– Use paths created in the previous phase to build expander on the

terminals

• Route original demands– Now use standard polylog(N)-approx for routing in expanders

Rao-Zhou Analysis

How to Attack General Case

• Obviously wrong approach

– Solve EDPwC in expanders

– Prove that every graph is an expander

So what else can we try?

• Divide and conquer?

– Not all graphs have good expansion…

– Can we partition any graph into subgraphs with good expansion?

– Yes!

– Use Räcke decomposition result

expander

expander

expander

Räcke Decompositions• Introduced for oblivious routing

– wl(S) = # edges in S between two level l clusters

– (Räcke) Can create log N levels s.t.

for all small S in level l cluster U

cap(S, U - S) ≥ wl +1 (S) / log N

Räcke Decompositions• Introduced for oblivious routing

– wl(S) = # edges in S between two level l clusters

– (Räcke) Can create log N levels s.t.

for all small S in level l cluster U

cap(S, U - S) ≥ wl +1 (S) / log N

• Contract level l+1 clusters• Get graph with good expansion

How to Attack General Case

• Different approach

– Prove that every graph is an expander-of-expanders

– Solve EDPwC in expanders-of-expanders ?

Use Räcke decomposition result

Expanders-of-Expanders

• How to route in expander of expanders

– Route recursively using expander routing?

– Seems difficult– At each level we can route a 1/log N

fraction of demands routed at higher level expander

expander

Uniform Decomposition

• But do we need to recurse across all levels?

– Suppose decomposition is uniform– For each cluster U either:

– all subclusters S have out(S) ≤ logp N or

– all subclusters S have out(S) ≥ logp N

small

large

Uniform Decomposition

• Critical clusters

– Cluster U is critical if:

– U is large

– all subclusters S of U are small

Three Step Routing

• Step 1– Shrink critical clusters to single nodes

– All cuts are now large

– Use Rao-Zhou

Three Step Routing

• Step 2– Shrink small clusters to single nodes

– Critical clusters now look like expanders

– Route in critical clusters using expander routing

Three Step Routing

• Step 3– Route across small clusters

– Small clusters have polylog( N ) terminals

– Using Rao-Zhou in small clusters gives poly( log log N ) congestion

Non-Uniform Result

• How can we handle non-uniform decompositions?– Basic idea: Ignore the non-uniform parts

– …

Open Problems

• Can we lower congestion by going further down the Räcke hierarchy?

• Can we improve the hardness example via a hierarchical construction?

Three Step Routing Revisited

• Step 1– Shrink critical clusters to single nodes

– All cuts are now large– Use Rao-Zhou (incl Karger sparsification lemma)

– i.e. partition edges in polylog(n) buckets using random sampling

– All cuts are preserved (up to polylog(n) factors)


• Step 2– Shrink small clusters to single nodes

– Critical clusters now look like expanders

– Route in critical clusters using expander routing


• Step 3– Route across small clusters

– Small clusters have polylog( N ) terminals

– Using Rao-Zhou in small clusters gives poly( log log N ) congestion

– But can we go further down the hierarchy to get better congestion?

Recursive Routing

• Small cluster routing– New defn for critical clusters based on poly(log log n) threshold

– Randomly partition edges in poly(log log n) buckets

– Problem!! - Could have poly(n) critical clusters

Small cluster outdegree polylog(n)

Outdegree poly( loglog(n))

New critical cluster

Hardness Revisited

• Could we improve hardness via a hierarchical example?– Original hardness result created on expander

– Hardness for congestion cannot be larger than log (diam) = log log N

Hardness Revisited

• Could we improve hardness via a hierarchical example?– Original hardness result created on expander

– Hardness for congestion cannot be larger than log (diam) = log log N

– Can we improve hardness by considering “hierarchical expander”

Expander

Expander

Expander

Hierarchical Decompositions

• What can uniform decompositions look like? • Key parameters

– l = Outdegree of level l cluster– Number of subclusters of level l cluster– Diameter of level l cluster when all subclusters are contracted


• Square mesh– l = Outdegree of level l cluster = 4 x 2l

– Number of subclusters of level l cluster = 4– Diameter of level l cluster = 2


• Hierarchy of Expanders– Suppose we want unique short path at each level– l = Outdegree of level l cluster = (l-1)d+1 / d– Number of subclusters of level l cluster = (l-1)d

– Diameter of level l cluster = d

– Polynomial relationship between outdegree and size of cluster– Shortest path between s and t remains ≤ log N

Expander

Expander

Expander


• Can we have the following in any natural decomposition?

• Can we handle this situation if it does occur?• Better sparsification result?

Outdegree of level l clusterl l-1 l-2 l-3

:2 1

Number of subclusters in level l clusterexp(l) exp(l-1 )exp(l-2 )exp(l-3 )

:exp(2 ) exp(1 )


• For hard examples– If outdegree of cluster is small, doesn’t make sense for the cluster

to be large– Either all terminals lie within the cluster– Or most of the cluster is not useful for routing

– Can we formalize this notion?

Small cluster outdegree polylog(n)

Thank you

Non-Uniform Result

• What could a non-uniform decomposition look like?– Set of critical clusters joined by trees

critical cluster

Non-Uniform Result


Not an acceptable cluster

Non-Uniform Result


Non-Uniform Result


Non-Uniform Result

• How do we deal with nodes outside critical cluster?– Replace each node in tree by an expander

Non-Uniform Result

• How do we deal with nodes outside critical cluster?– Use expanders to route on tree using low congestion paths

Non-Uniform Result

– Use paths to join up critical clusters

– Key property #1: all tree nodes are gone

– Key property #2: all cuts are preserved (more or less)

Use analysis for uniform decompositions!!

Non-Uniform Result

• But what about general non-uniform decompositions?– H = nodes that aren’t in a critical cluster

H

critical cluster

Non-Uniform Result

• But what about general non-uniform decompositions?– H = nodes that aren’t in a critical cluster

H

Non-Uniform Result

– Use a Räcke lemma to find small clusters around outside of H

– Let H’ be the subset of H that is not in any of these clusters

– |out( H’ )| ≤ |out( H )| / 2

– Can do this at most log(N) times

H H’

Non-Uniform Result

– Repeat log N times

– Use small clusters to grow paths of logarithmic length that connect up critical clusters

H

critical cluster

Non-Uniform Result

– So now we have critical clusters joined by paths of logarithmic length (just as in the “tree” case)

– We are done…

H

All Rights Reserved © Alcatel-Lucent 2006, ##### Matthew Andrews, Alcatel-Lucent Bell Labs Princeton Approximation Workshop June 15, 2011 Edge-Disjoint.

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