Polynomial Bounds for the Grid-Minor Theorem

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

2. 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

3. 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 treewidthof G is large, then G contains a large grid minor

4. Treewidth General Graphs Trees

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

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

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

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

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

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

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

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

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

14. Well-Linkedness

15. 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.

16. 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.

17. Treewidth Large-Treewidth Graphs Small-Treewidth Graphs Trees

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

19. Grid-Minor Theorem[Robertson, Seymour] If the treewidthof 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

20. Minors by Embedding

21. Minors by Embedding

22. Grid-Minor Theorem[Robertson, Seymour] If the treewidthof 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 • …

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

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

25. Grid-Minor Theorem If the treewidthof 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)?

26. Grid-Minor Theorem If the treewidthof 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:

27. Grid-Minor Theorem If the treewidthof 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]

28. Path-of-Sets System

29. A Path-of-Sets System C1 C2 C3 … Ch … • Each Ci 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

30. A Path-of-Sets System C1 C2 C3 … Ch … h • Each Ci 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

31. A Path-of-Sets System C1 C2 C3 … Ch … Ci • The interface vertices are well-linked inside Ci Interface vertex

32. A Path-of-Sets System C1 C2 C3 … Ch … Ci • The interface vertices are well-linked inside Ci

33. A Path-of-Sets System C1 C2 C3 … Ch Ci • The interface vertices are well-linked inside Ci

34. A Path-of-Sets System C1 C2 C3 … Ch Ci • The interface vertices are well-linked inside Ci

35. A Path-of-Sets System C1 C2 C3 … Ch Ci • The interface vertices are well-linked inside Ci

36. A Path-of-Sets System C1 C2 C3 … Ch … 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.

37. From Path-of-Sets System to Grid Minor

38. Building the Grid

39. Building the Grid

40. Building the Grid

41. Building the Grid

42. Building the Grid

43. Building the Grid C1 C2 C3 … Ch • C1 • C4 • C2 P1 P2 P3 • C3 Ph … …

44. Direct vs Indirect Path Direct path Indirect path

45. Building the Grid C1 C1 C2 C2 C3 C3 … … Ch Ch • C1 For each Ci, we’ll be looking for a direct path connecting some consecutive pair of horizontal paths • C4 • C2 P1 P2 P3 • C3 Ph … …

46. Routing Inside Clusters Ci P1 P2 P3 P4

47. Routing Inside Clusters Ci P1 P4 P1 P2 P2 P3 P3 P4 Path graph Hi for Ci

48. Routing Inside Clusters Ci P1 P4 P1 P2 P2 P3 P3 P4 Path graph Hi for Ci

49. Routing Inside Clusters Ci P1 P4 P1 P2 P2 P3 P3 P4 Path graph Hi for Ci

50. Routing Inside Clusters P1 Good scenario: The path graph for all Ci contains the same path “Bad” scenario: P2 P1 P3 P4 P2 P3 P4