Approximation Algorithms for Graph Routing Problems Julia Chuzhoy Toyota Technological Institute at Chicago

Routing Problems Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible; minimize edge congestion.

Routing Problems Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible; minimize edge congestion.

Routing Problems Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible; minimize edge congestion. Edge congestion: 2

Routing Problems Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible; minimize edge congestion. n – number of graph vertices m – number of edges k – number of demand pairs terminals – vertices participating in the demand pairs n – number of graph vertices m – number of edges k – number of demand pairs terminals – vertices participating in the demand pairs

Routing Problems Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible; minimize edge congestion. 3 pairs with congestion 2

Routing Problems Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible; minimize edge congestion. 3 pairs with congestion 2 2 pairs with congestion 1 3 pairs with congestion 2 2 pairs with congestion 1

Congestion Minimization

Route all demand pairs Minimize maximum edge congestion solution value: 2

LP-relaxation Rounding Algorithm [Raghavan, Thompson ‘87] Every pair (s i,t i ) chooses a path P w. probability f(P) Max edge congestion O(c  log n/log log n) w.h.p.

Congestion Minimization -approximation [Raghavan, Thompson ‘87] Directed graphs: -hard to approximate [Andrews, Zhang ‘06], [C, Guruswami, Khanna, Talwar ‘07] Undirected graphs: -hard to approximate [Andrews, Zhang ‘07]

Open Problem 1 Can we close the gap for undirected graphs? What is the integrality gap of the LP? What if we only need to connect a constant fraction of the demand pairs?

Edge-Disjoint Paths Problem No congestion allowed Route maximum number of the demand pairs Solution value: 2

Edge Disjoint Paths (EDP) For directed graphs NP-hard even with only two demand pairs [Fortune, Hopcroft, Wyllie '80] When k is constant, can be solved efficiently in undirected graphs [Robertson, Seymour ‘90] NP-hard when k is part of input [Karp ’72]

LP-relaxation

Rounding Algorithm [Kolliopoulos, Stein ‘98] Find shortest path P with non-zero flow, connecting any demand pair. Add P to the solution and delete all flow that uses edges of P. Analysis If the length of P is less than – at most flow is deleted. If the length of P is more than – at most flow remains. -approximation

Can We Do Better? Directed graphs: EDP is -hard to approximate for any [Guruswami, Khanna, Rajaraman, Shepherd, Yannakakis ‘99] Undirected graphs : -approximation algorithm [Chekuri, Khanna, Shepherd ’06]. -hardness of approximation for any [Andrews, Zhang ‘05], [Andrews, C, Guruswami, Khanna, Talwar, Zhang ’10] LP integrality gap: [Garg, Vazirani, Yannakakis ‘93]

Integrality Gap Example [Garg, Vazirani, Yannakakis ‘93] s1s1 s2s2 sksk … t1t1 t2t2 tktk … fractional: k/2 integral: 1 k= fractional: k/2 integral: 1 k=

A Brick-Wall Graph s1s1 s2s2 sksk … tktk t1t1 t2t2 …

Open Problem 2 Close the gap for undirected EDP planar graphs? – constant-factor approximation with congestion 2 [Chekuri, Khanna, Shepherd ‘04], [Chekuri, Khanna, Shepherd ‘06], [Seguin-Charbonneau, Shepherd ‘11] – O(log n)-approximation for Eulerian or 4-edge connected planar graphs [Kawarabayashi, Kobayashi ’10] – constant approximation for grid-like graphs [Kleinberg, Tardos ’95]

Open Problem 2 Close the gap for undirected EDP planar graphs? better algorithms for brick-wall graphs?

Positive Results Moderately connected graphs: If global min-cut is Ω(log 5 n), there is a polylog-approximation [Rao, Zhou ‘10] Expander graphs In a strong enough constant-degree expander, any demand set on vertices can be routed on edge-disjoint paths [Frieze ‘00] …

Edge Disjoint Paths (EDP) Route maximum number of pairs on edge-disjoint paths -approximation matching integrality gap -hardness Congestion Minimization Route all pairs; minimize congestion -approximation -hardness EDP with Congestion (EDPwC) A factor- approximation algorithm with congestion c routes. demand pairs with congestion at most c. optimum number of pairs with no congestion allowed

EDPwC Congestion O(log n/log log n): constant approximation [Raghavan, Thompson ’87] -approximation with congestion c [Azar, Regev ’01], [Baveja, Srinivasan ’00], [Kolliopoulos, Stein ‘04] Directed graphs: -hardness for any congestion c [C, Guruswami, Khanna, Talwar ’06]

EDPwC Congestion O(log n/log log n): constant approximation [Raghavan, Thompson ’87] -approximation with congestion c [Azar, Regev ’01], [Baveja, Srinivasan ’00], [Kolliopoulos, Stein ‘04] polylog(n)-approximation with congestion poly(log log n) [Andrews ‘10] Congestion 2: -approximation [Kawarabayashi, Kobayashi ’11] polylog(k)-approximation with congestion 14 [C, ‘11] polylog(k)-approximation with congestion 2 [C, Li, ‘12]

Integrality Gaps for EDPwC s1s1 s2s2 sksk … t1t1 t2t2 tktk … Congestion 1: integrality gap [Garg, Vazirani, Yannakakis ‘93] Congestion c: integrality gap [Andrews, C, Guruswami, Khanna, Talwar, Zhang ‘10] -hard to approximate with congestion c for any

Edge Disjoint Paths (EDP) Route maximum number of pairs on edge-disjoint paths -approximation matching integrality gap -hardness Congestion Minimization Route all pairs; minimize congestion -approximation -hardness EDP with Congestion (EDPwC) polylog(k)-approximation with congestion 2 -hardness with congestion c

Another View: Reducing Congestion Suppose we have a “bad” solution, where X pairs are routed with congestion C. Then we can route X/(C polylog k) pairs with congestion 2! But if we want congestion 1, may only be able to route pairs, even if C=2.

A Polylogarithmic Approximation with Constant Congestion

Well-Linkedness [Robertson,Seymour], [Chekuri, Khanna, Shepherd], [Raecke] Graph G is well-linked for the set T of terminals, iff for any partition (A,B) of V(G),

Well-Linkedness [Robertson,Seymour], [Chekuri, Khanna, Shepherd], [Raecke] Graph G is well-linked for the set T of terminals, iff for any partition (A,B) of V(G),

Pre-Processing Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Theorem [Chekuri, Khanna Shepherd ‘04] Can efficiently partition G into disjoint subgraphs G 1,…, G r, such that: For each induced sub-problem G i the terminals are well-linked Total fractional solution value for all induced sub-problems is “Enough” to solve the problem on well- linked instances.

1.Embed an expander on a subset of terminals into G. – expander vertices terminals – expander edges paths in G 2.Route a subset of the demand pairs in the expander High-Level Plan [CKS ‘04] Embedding congestion: max load on any edge of G Crossbar

Goal Embed an expander over a subset of terminals into G. Include polylog(k)-fraction of the terminals congestion 2

Cut-Matching Game [Khandekar, Rao, Vazirani ’06] Cut Player: wants to build an expander Matching Player: wants to delay its construction There is a strategy for cut player, s.t. after O(log 2 n) iterations, we get an expander!

Embedding Expander into Graph

After O(log 2 k) iterations, we get an expander embedded into G. Problem: congestion Ω(log 2 k)

Solution? Idea [Rao Zhou ‘06]: Split G into graphs G 1,…,G h using algorithm of [Karger ’93] – V(G i )=V(G) for all i – Every edge of G belongs to at most one G i – Each G i well-linked for the terminals – h=O(log 2 k) Run the cut-matching game. Use G i to route flow in iteration i. Problem: Can only do it if min-cut in G is Ω(log 5 n) [Andrews ‘10] adapted this to general graphs, with congestion poly(log log n)

Getting a Constant Congestion

Embedding an Expander into G Expander vertex connected component in G containing the terminal Expander edge path connecting some pair of vertices in the two components An edge of G belongs to at most 2 of the components/paths.

Embedding an Expander into G Routing on vertex-disjoint paths in X gives a congestion-2 routing in G!

Expander vertex connected component in G containing the terminal Expander edge path connecting some pair of vertices in the two components An edge of G belongs to at most 2 of the components/paths. Embedding an Expander into G

Families of Good Vertex Sets

Good Vertex Subset S is a good vertex subset iff: S contains no terminals k/polylog k red edges S is well-linked for the red edges

A Good Family of Vertex Subsets O(log 2 k) disjoint good vertex subsets k/polylog k trees −each edge of G participates in at most 2 trees −Each tree T i spans a distinct terminal t i and a distinct red edge adjacent to S j for each j.

Embedding an Expander

Expander vertex the tree spanning the terminal Expander edges: via the cut-matching game of [KRV]

Embedding an Expander

After O(log 2 k) iterations, we obtain an expander embedded into G with congestion 2.…

Algorithm for EDPwC Find a good family of vertex subsets Embed an expander into G Find vertex-disjoint routing in the expander Transform into routing in G

Open Problem 3 A cleaner algorithm for polylog(k) approximation with congestion 2? Better power of polylog? What if we need to route almost all demand pairs (a constant fraction)?

Routing on Vertex-Disjoint Paths Generalizes EDP Connections to Graph Minor theory No congestion: -approximation [Kolliopoulos, Stein ‘98] -lower bound on LP integrality gap [Garg, Vazirani, Yannakakis ‘97] Routing with congestion: O(poly log k)-approximation with constant congestion [Chekuri, Ene ‘12]

Better algorithms/lower bounds for Vertex- Disjoint Paths? Open Problem 4

Better algorithms/lower bounds for Vertex- Disjoint Paths? Vertex-disjoint paths in grid graphs Open Problem 4

s1s1 t1t1 s2s2 t2t2 s3s3 t3t3 Better algorithms/lower bounds for Vertex- Disjoint Paths? Vertex-disjoint paths in grid graphs Open Problem 4

Thank you!