All Rights Reserved © Alcatel-Lucent 2006, ##### Matthew Andrews Show-and-Tell April 20, 2010 Edge Disjoint Paths via Räcke Decompositions

Edge Disjoint Paths (EDP) Input –Graph G (M edges, N nodes); –A set of demands, (s i, t i ); Output –A subset of demands routed on edge-disjoint paths; –Maximize such a subset s t t s OPT = 1

Edge Disjoint Paths (EDP) Input –Graph G (M edges, N nodes); –A set of demands, (s i, t i ); Output –A subset of demands routed on edge-disjoint paths; –Maximize such a subset s t t s OPT = 1 Hard problem One of the first!! (Karp’s list)

Congestion Minimization Input –Graph G (M edges, N nodes); –A set of demands, (s i, t i ); 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, (s i, t i ); –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 –N 1/c - approx with congestion c –Azar-Regev, Baveja-Srinivasan, Kolliopoulos-Stein Negative –No log (1-  )/(c+1) N - approx with congestion c –A-Chuzhoy-Guruswami-Khanna-Talwar-Zhang Open Question: Is there a polylog(N)-approx with constant congestion?

Known Results (Undirected Graphs) Positive –N 1/c - approx with congestion c –Azar-Regev, Baveja-Srinivasan, Kolliopoulos-Stein Negative –No log (1-  )/(c+1) N - approx with congestion c –A-Chuzhoy-Guruswami-Khanna-Talwar-Zhang Open Question: Is there a polylog(N)-approx with constant congestion? Yes for planar graphs, “all-or-nothing” flow (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

Known Results (Undirected Graphs) Positive –N 1/c - approx with congestion c –Azar-Regev, Baveja-Srinivasan, Kolliopoulos-Stein Negative –No log (1-  )/(c+1) N - approx with congestion c –A-Chuzhoy-Guruswami-Khanna-Talwar-Zhang Rao-Zhou conjecture: There is a polylog(N)-approx with O(log log N) congestion

Results (Undirected Graphs) Positive –N 1/c - approx with congestion c –Azar-Regev, Baveja-Srinivasan, Kolliopoulos-Stein Negative –No log (1-  )/(c+1) N - approx with congestion c –A-Chuzhoy-Guruswami-Khanna-Talwar-Zhang This talk: There is a polylog(N)-approx with poly(log log N) congestion

Results (Undirected Graphs) Positive –N 1/c - approx with congestion c –Azar-Regev, Baveja-Srinivasan, Kolliopoulos-Stein Negative –No log (1-  )/(c+1) N - approx with congestion c –A-Chuzhoy-Guruswami-Khanna-Talwar-Zhang This talk: There is a poly log(N)-approx with poly (log log N) congestion

Results Positive –N 1/c - approx with congestion c –Azar-Regev, Baveja-Srinivasan, Kolliopoulos-Stein Negative –No log (1-  )/(c+1) N - approx with congestion c –A-Chuzhoy-Guruswami-Khanna-Talwar-Zhang This talk: There is a log 61 (N)-approx with (log log N) 6 congestion

Talk Outline Describe some previous work on EDP Describe Rao-Zhou and Räcke −Rao-Zhou: EDP in graphs with large min-cut −Räcke: hierarchical graph decompositions Merge previous analyses to get EDPwC result

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), answer to open question is “yes” –Why? –Use randomized rounding. Each edge is dependent on polylog(N) other edges –Apply Lovasz Local Lemma Open Question: Is there a polylog(N)-approx with constant congestion? s t t s 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 –Answer to open question in expanders is “yes” –e.g. Broder-Frieze-Upfal, Kolman-Scheideler –Even better, can connect polylog(N) fraction of ANY set of terminals using constant congestion Why? –Using random walks, can connect any pair of terminals with paths of polylog(N) length –Use known result #1

Rao-Zhou Known result #3 –(Rao-Zhou) Answer to open question is “yes” if min-cut in graph has size polylog(k) ---- ( k = # source-demand pairs) Open Question: Is there a polylog(N)-approx with constant congestion?

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) 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

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

Räcke Decompositions Introduced for oblivious routing –w l (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) ≥ w l +1 (S) / log N

Räcke Decompositions Introduced for oblivious routing –w l (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) ≥ w l +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

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) ≤ log p N or –all subclusters S have out(S) ≥ log p 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 What could a non-uniform decomposition look like? –Set of critical clusters joined by trees critical cluster

Non-Uniform Result What could a non-uniform decomposition look like? –Set of critical clusters joined by trees Not an acceptable cluster

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

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

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 Racke 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

Thank you