Polynomial Bounds for the Grid-Minor Theorem Julia Chuzhoy Toyota Technological Institute at Chicago Chandra Chekuri University of Illinois at Urbana-Champaign

Grid Minor Theorem (Excluded Grid Theorem) [Robertson, Seymour ‘86] Graph Minor Theory [Robertson – Seymour] – Wagner’s conjecture: any infinite sequence of finite graphs contains two graphs G,G’ where G is a minor of G’ – Grid-Minor Theorem: if the treewidth of G is large, then G contains a large grid minor

Grid Minor Theorem (Excluded Grid Theorem) [Robertson, Seymour ‘86] Graph Minor Theory [Robertson – Seymour] – Wagner’s conjecture: any infinite sequence of finite graphs contains two graphs G,G’ where G is a minor of G’ – Grid-Minor Theorem: if the treewidth of G is large, then G contains a large grid minor

Treewidth Trees General Graphs

Tree Decomposition d e b c a f g h Example from Bodlaender’s talk

g h g a f a f c c d e Tree Decomposition d e b c a f g h b c a Example from Bodlaender’s talk

c d e Tree Decomposition d e b c a f g h a f g h c g a f b c a Example from Bodlaender’s talk

c d e Tree Decomposition d e b c a f g h a f g h c g a f b c a Example from Bodlaender’s talk

c d e Tree Decomposition d e b c a f g h a f g h c g a f b c a Example from Bodlaender’s talk

c d e Tree Decomposition d e b c a f g h a f g h c g a f b c a Example from Bodlaender’s talk

c d e Tree Decomposition d e b c a f g h a f g h c g a f b c a Example from Bodlaender’s talk

c d e Tree Decomposition d e b c a f g h a f g h c g a f b c a Example from Bodlaender’s talk Decomposition width = max # of vertices in a bag -1 Treewidth: min width of any decomposition

Treewidth of Some Graphs Tree: 1 Cycle: 2 (√n×√n)-grid: √n n-vertex expander: Ω(n)

Well-Linkedness

A set T of vertices is well-linked in G iff for any two equal- sized subsets A,B of T, we can connect A to B with |A| disjoint paths.

Treewidth and Well-Linkedness Thm. Let k be the maximum size of any well- linked set of vertices in G. Then: k≤treewidth(G)≤4k.

Treewidth Trees Small-Treewidth Graphs Large-Treewidth Graphs

Grid-Minor Theorem [Robertson, Seymour] If the treewidth of G is large, then it contains a large grid minor.

Grid-Minor Theorem [Robertson, Seymour] If the treewidth of G is large, then it contains a large grid minor. We can obtain the grid from G by a sequence of edge-deletion and edge-contraction operations a size-4 grid

Minors by Embedding

Grid-Minor Theorem [Robertson, Seymour] If the treewidth of G is large, then it contains a large grid minor, so: G contains many disjoint cycles G contains many disjoint cycles of length 0 mod m G contains a convenient routing structure The size of the vertex cover in G is large …

Applications Fixed parameter tractability Erdos-Posa type results Graph minor theory …

If the treewidth of G is large, then it contains a large grid minor. Grid-Minor Theorem

If the treewidth of G is k, then it contains a grid minor of size f(k). Easy to see that [Robertson, Seymour ‘94]: Conjecture [Robertson, Seymour ‘94]: How large is f(k)?

Grid-Minor Theorem If the treewidth of G is k, then it contains a grid minor of size f(k). [Robertson, Seymour, Thomas ‘89] : [Diestel, Gorbunov, Jensen, Thomassen ‘99] – simpler proof [Kawarabayashi, Kobayashi ‘12], [Leaf, Seymour ‘12]: This talk:

Grid-Minor Theorem If the treewidth of G is k, then it contains a grid minor of size f(k). In some families of graphs f(k)=Ω(k) – Planar graphs [Robertson, Seymour, Thomas ‘94] – Bounded genus graphs [Demaine, Fomin, Hajiaghayi, Thilikos ‘05] – Graphs excluding a fixed minor [Demaine, Hajiaghayi ‘08]

Path-of-Sets System

A Path-of-Sets System C1C1 C2C2 C3C3 …ChCh Each C i is a connected cluster The clusters are disjoint Every consecutive pair of clusters connected by h paths All blue paths are disjoint from each other and internally disjoint from the clusters …

A Path-of-Sets System C1C1 C2C2 C3C3 …ChCh Each C i is a connected cluster The clusters are disjoint Every consecutive pair of clusters connected by h paths All blue paths are disjoint from each other and internally disjoint from the clusters h …

A Path-of-Sets System C1C1 C2C2 C3C3 …ChCh … CiCi Interface vertex The interface vertices are well-linked inside C i

A Path-of-Sets System C1C1 C2C2 C3C3 …ChCh … CiCi The interface vertices are well-linked inside C i

A Path-of-Sets System C1C1 C2C2 C3C3 …ChCh CiCi The interface vertices are well-linked inside C i

A Path-of-Sets System C1C1 C2C2 C3C3 …ChCh The interface vertices are well-linked inside C i CiCi

A Path-of-Sets System C1C1 C2C2 C3C3 …ChCh The interface vertices are well-linked inside C i CiCi

A Path-of-Sets System C1C1 C2C2 C3C3 …ChCh h … Thm [Leaf, Seymour ‘12]: Given a path-of-sets system, we can efficiently find a grid minor of size Ω(√h). Corollary: enough to find a path-of-sets system with h=poly(k), where k is the treewidth.

From Path-of-Sets System to Grid Minor

Building the Grid

C1C1 C1C1 C2C2 C2C2 C3C3 C3C3 C4C4 C4C4 C1C1 C2C2 C3C3 …ChCh P1P1 P2P2 P3P3 PhPh … …

Direct vs Indirect Path Direct path Indirect path

Building the Grid C1C1 C2C2 C3C3 …ChCh C1C1 C1C1 C2C2 C2C2 C3C3 C3C3 C4C4 C4C4 For each C i, we’ll be looking for a direct path connecting some consecutive pair of horizontal paths C1C1 C2C2 C3C3 …ChCh P1P1 P2P2 P3P3 PhPh … …

Routing Inside Clusters CiCi P1P1 P2P2 P3P3 P4P4

CiCi P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 Path graph H i for C i

Routing Inside Clusters CiCi P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 Path graph H i for C i

Routing Inside Clusters CiCi P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 Path graph H i for C i

Routing Inside Clusters P1P1 P2P2 P3P3 P4P4 Good scenario: The path graph for all C i contains the same path P1P1 P2P2 P3P3 P4P4 “Bad” scenario:

Routing Inside Clusters CiCi P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 Path graph H i for C i

Inside the Super-Clusters Thm: for any n-vertex graph G, Either there is a tree in G with Ω(√n) leaves Or there is a 2-path in G of length Ω(√n)

Inside the Super-Clusters Thm: for any n-vertex graph G, Either there is a tree in G with Ω(√n) leaves Or there is a 2-path in G of length Ω(√n)

Routing Inside Clusters CiCi P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 Path graph H i for C i Cluster C i is good if H i has a tree with √h leaves. Assume all clusters are good.

Routing Inside Clusters CiCi P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 Path graph H i for C i

Routing Inside Clusters CiCi P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 Path graph H i for C i

Routing Inside Clusters CiCi P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 Path graph H i for C i We say that C i chooses the paths corresponding to the leaves of the tree.

Routing Inside Clusters …

…

…

If r is large enough, then some choice of √h will repeat h times. … r

Routing Inside Clusters

Re-connect the paths via even-indexed clusters, so all odd-indexed clusters choose the same paths!

Completing the Proof Super- cluster

Completing the Proof For each super-cluster S i : Either build a large grid minor inside S i Or show that S i is a good cluster

Inside the Super-Clusters P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 H: path-graph for the super- cluster

Inside the Super-Clusters P1P1 P2P2 P3P3 P4P4 P1P1 P2P2 P3P3 P4P4 H: path-graph for the super- cluster Either H contains a tree with many leaves Or it contains a long 2-path  Can build a grid-minor directly Either H contains a tree with many leaves Or it contains a long 2-path  Can build a grid-minor directly P1P1 P2P2 P3P3 P4P4 H1H1 P1P1 P2P2 P3P3 P4P4 H2H2 P1P1 P2P2 P3P3 P4P4 H4H4 P1P1 P2P2 P3P3 P4P4 H3H3

Inside the Super-Clusters P1P1 P2P2 P3P3 P4P4 H: path-graph for the super- cluster P1P1 P2P2 P3P3 P4P4 H1H1 P1P1 P2P2 P3P3 P4P4 H2H2 P1P1 P2P2 P3P3 P4P4 H4H4 P1P1 P2P2 P3P3 P4P4 H3H3 v1v1 v2v2 v3v3 v4v4 …v √h Want to show: this path appears in all H i ’s Will show: large sub-path appears in half the H i ’s Want to show: this path appears in all H i ’s Will show: large sub-path appears in half the H i ’s

Inside the Super-Clusters P1P1 P2P2 P3P3 P4P4 H: path-graph for the super- cluster P1P1 P2P2 P3P3 P4P4 H1H1 v1v1 v2v2 v3v3 v4v4 …v √h

Inside the Super-Clusters P1P1 P2P2 P3P3 P4P4 H: path-graph for the super- cluster P1P1 P2P2 P3P3 P4P4 H1H1 v1v1 v2v2 v3v3 v4v4 …v √h

Inside the Super-Clusters P1P1 P2P2 P3P3 P4P4 H: path-graph for the super- cluster P1P1 P2P2 P3P3 P4P4 H1H1 P1P1 P2P2 P3P3 P4P4 H2H2 P1P1 P2P2 P3P3 P4P4 H4H4 P1P1 P2P2 P3P3 P4P4 H3H3 v1v1 v2v2 v3v3 v4v4 …v √h

Completing the Proof For each super-cluster S i : Either build a large grid minor inside S i Or show that S i is a good cluster

Finding the Path-of-Sets System C1C1 C2C2 C3C3 …ChCh …

Routing Problems

Node-Disjoint Paths (NDP) Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible via node- disjoint paths

Node-Disjoint Paths (NDP) Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible via node- disjoint paths

Node-Disjoint Paths (NDP) Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible via node- disjoint paths Solution value: 2

Edge-Disjoint Paths (EDP) Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible via edge- disjoint paths

Edge-Disjoint Paths (EDP) Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible via edge- disjoint paths Solution value: 3

Edge-Disjoint Paths (EDP) Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible via edge- disjoint paths NDP is more general than EDP

Edge-Disjoint Paths (EDP) Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Route as many pairs as possible via edge- disjoint paths n – number of graph vertices k – number of demand pairs terminals – vertices participating in the demand pairs n – number of graph vertices k – number of demand pairs terminals – vertices participating in the demand pairs

EDP and NDP Efficient algorithm when k is constant [Robertson, Seymour ‘90]. – running time: f(k)  n 2 [Kawarabayashi,Kobayashi, Reed] General k: both problems are NP-hard [Karp ’72] An α-approximation algorithm: efficient algorithm always produces solutions of value at least OPT/α. An α-approximation algorithm: efficient algorithm always produces solutions of value at least OPT/α.

Approximation Algorithm [Kolliopoulos, Stein ‘98] While there is a path P connecting any demand pair that has not been routed yet: Add such a path of smallest length to the solution (Delete from OPT all paths sharing vertices with P or routing the same demand pair) Analysis If the length of P is less than – at most paths are deleted from OPT. If the length of P is more than – at most paths remain in OPT.

Approximation Algorithm [Kolliopoulos, Stein ‘98] While there is a path P connecting any demand pair that has not been routed yet: Add such a path of smallest length to the solution (Delete from OPT all paths sharing vertices with P or routing the same demand pair) Analysis If the length of P is less than – at most paths are deleted from OPT. If the length of P is more than – at most paths remain in OPT. -approximation This algorithm gives an – approximation for EDP. An -approximation is known [Chekuri, Khanna, Shepherd ’06]. This algorithm gives an – approximation for EDP. An -approximation is known [Chekuri, Khanna, Shepherd ’06].

Approximation Status of NDP -approximation algorithm – Even on planar graphs – Even on grid graphs -hardness of approximation for any[Andrews, Zhang ‘05], [Andrews, C, Guruswami, Khanna, Talwar, Zhang ’10]

Open Problem: NDP on a Grid Graph

s1s1 t1t1 s2s2 t2t2 s3s3 t3t3

-approximation algorithm [Chekuri, Khanna, Shepherd ’06]. The problem is NP-hard Ongoing work: O(n 1/4 )-approximation [builds on Aggarwal, Kleinberg, Williamson ‘96] Hard to approximate up to some constant c.

Open Problem: EDP on Wall Graphs

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] polylog(n)-approximation with congestion poly(log log n) [Andrews ‘10] polylog(k)-approximation with congestion 14 [C, ‘11] polylog(k)-approximation with congestion 2 [C, Li, ‘12] polylog(k)-approximation with constant congestion for NDP [Chekuri, Ene ’13]

Edge-Disjoint Paths Input: Graph G, source-sink pairs (s 1,t 1 ),…,(s k,t k ). Goal: Connect as many pairs as possible by edge-disjoint paths. An instance is well-linked iff the set of all terminals is well-linked in G. Theorem [Chekuri, Khanna Shepherd ‘04]: an α - approximation algorithm on well-linked instances gives an O(α log 2 k)-approximation on any instance. terminals

If an instance is well-linked, its treewidth is Ω(k) If the treewidth of G is k, can find a well-linked set of size Algorithms for Edge-Disjoint Paths well-linked instance large crossbar find the routing graph of treewidth k “similar” to path-of-sets system

Crossbar Number of clusters poly(log k), not poly(k) The paths are not disjoint from each other and from the clusters, but cause a constant edge congestion Want: Path-of-sets system Can get: Tree-of-sets system … degree-3 tree ✔ ✔

Tree-of-Sets System A degree-3 tree with h vertices Every vertex a connected cluster of G Every edge – a collection of h paths in G – the blue paths are node-disjoint from each other and internally disjoint from the clusters For each cluster, its interface is well-linked. h=k ε If the tree has height h 1/3 – done Otherwise it has h 1/3 leaves Will build a path-of-sets system on a subset of h 1/3 leaves Assumption: the tree has h 1/3 leaves

High-Level Idea

Stage 1: connect every leaf to the root by many disjoint paths Stage 2: exploit these paths to build a path- of-sets system

Stage 1 h 1/3 leaves h parallel blue edges each leaf gets h 3/4 green paths h 1/3 leaves h parallel blue edges each leaf gets h 3/4 green paths

Stage 1 h 1/3 leaves h parallel blue edges each leaf gets h 3/4 green paths h 1/3 leaves h parallel blue edges each leaf gets h 3/4 green paths

Stage 1 h 1/3 leaves h parallel blue edges each leaf gets h 3/4 green paths h 1/3 leaves h parallel blue edges each leaf gets h 3/4 green paths

Stage 1 h 1/3 leaves h parallel blue edges each leaf gets h 2/3 green paths h 1/3 leaves h parallel blue edges each leaf gets h 2/3 green paths

Stage 2 Every leaf receives h 2/3 flow units from the root Will exploit these flows to build a path-of-sets system Process the tree from top to bottom

Stage 2

AB

AB

A B C D X R

A B A’ B’ C D C’ D’ X R

Stage 2 A B A’ B’ C D C’ D’ h 1/3 blue paths intersect at most h 1/3 green paths from each set h 1/3 blue paths intersect at most h 1/3 green paths from each set R X

Stage 2 A B A’ B’ C D C’ D’ B’ C C’ h 1/3 leaves each leaf had h 2/3 green paths want h 1/3 parallel paths in path-of-sets system tree height ≤ h 1/3 h 1/3 leaves each leaf had h 2/3 green paths want h 1/3 parallel paths in path-of-sets system tree height ≤ h 1/3 R X

Proof Summary 1.Path-of-sets system gives a large grid minor [Leaf, Seymour ‘12] 2.If G has large treewidth, can build a large tree-of-sets system: extension of [C ‘11], [C, Li ‘12], [Chekuri, Ene ‘12] 3.Can build a path-of-sets system from a tree- of-sets system polylog(k)-approximation for Node-Disjoint Paths with congestion 2

Bypassing the Grid-Minor Theorem?

Large-Treewidth Graph Decomposition [C, Chekuri ‘12] treewidth ≥ r G Treewidth k G Treewidth k h

Example of Use: Feedback Vertex Set Feedback Vertex Set: given a graph G, select a min-cardinality subset U of vertices, such that G\U has no cycles. k: size of feedback vertex set Want: a fixed-parameter tractable algorithm, with running time f(k)  poly(n).

The Algorithm If treewidth of G at most g(k) – dynamic programming on the tree decomposition – running time: otherwise: G contains a grid minor of size, so feedback vertex set value more than k. Can choose, running time. What is g(k)? Typical use of grid-minor theorem in fixed parameter tractability algorithms. Bi-dimentionality theory Typical use of grid-minor theorem in fixed parameter tractability algorithms. Bi-dimentionality theory

If treewidth of G at most g(k) – dynamic programming on the tree decomposition – running time: otherwise: G contains a grid minor of size, so feedback vertex set value more than k. Can choose, running time. The Algorithm What is g(k)?

Large-Treewidth Graph Decomposition treewidth ≥ 2 G G k+1 feedback vertex set value at least k+1

If treewidth of G at most g(k) – dynamic programming on the tree decomposition – running time: otherwise: G contains a grid minor of size, so feedback vertex set value more than k. Can choose, running time. The Algorithm What is g(k)?

Conclusion First polynomial bound on grid minor size,, Best current negative result: Better upper/lower bounds? Better/simpler constructions of path-of-sets or tree-of-sets systems?

More Open Questions Approximability of NDP/EDP: – general graphs – planar graphs – grid/wall graphs Congestion minimization – -approximation [Raghavan, Thompson ‘87] – -hard to approximate [Andrews, Zhang ‘07] Many more… Thank you!