1 / 38

On Tractable Parameterizations of Graph Isomorphism

On Tractable Parameterizations of Graph Isomorphism. Adam Bouland, Anuj Dawar and Eryk Kopczyński. G. H. Is ?. G 1 G 2. What is the parameterized complexity of Graph Isomorphism?. Size of smallest excluded minor. Tree-Width. Genus. Crossing Number. Path-Width.

fedora
Télécharger la présentation

On Tractable Parameterizations of Graph Isomorphism

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. On Tractable Parameterizations of Graph Isomorphism Adam Bouland, Anuj Dawar and Eryk Kopczyński

  2. G H Is ? G1 G2

  3. What is the parameterized complexity of Graph Isomorphism?

  4. Size of smallest excluded minor Tree-Width Genus Crossing Number Path-Width Tree-Depth Max Leaf Number Vertex Cover Number

  5. Size of smallest excluded minor XP Tree-Width Genus nf(k) Crossing Number Path-Width Tree-Depth Max Leaf Number Vertex Cover Number

  6. Size of smallest excluded minor ? FPT ? ? Tree-Width Genus f(k)nO(1) ? ? Crossing Number Path-Width + Others ? Tree-Depth Max Leaf Number Vertex Cover Number

  7. Size of smallest excluded minor ? FPT ? ? Tree-Width Genus ? ? Crossing Number Path-Width Tree-Depth Max Leaf Number Vertex Cover Number

  8. Size of smallest excluded minor ? FPT ? ? Tree-Width Genus ? ? Crossing Number Path-Width Generalized Tree-Depth Tree-Depth Max Leaf Number Vertex Cover Number

  9. Why tree-depth? Theorem [Elberfeld Grohe Tantau 2012]: FO=MSO on a class of graphs C iff C has bounded tree-depth Game definition – similar to path-width Matrix factorization

  10. Tree-Depth: 2 definitions Rooted Forest “Closure” of Forest G has td(G)<=d iff G is a subgraph of the closure of a forest of depth d.

  11. Proof Outline • Decomposition • Modify tree isomorphism algorithm • Bound # vertices which can serve as root of decomposition

  12. Proof Outline • Decomposition • Bound # vertices which can serve as root of decomposition • Modify tree isomorphism algorithm

  13. Tree-Depth: 2 definitions d cops 1 robber Cop player wins if a cop lands on the robber

  14. Tree-Depth: 2 definitions d cops 1 robber

  15. Tree-Depth: 2 definitions d cops 1 robber

  16. Tree-Depth: 2 definitions d cops 1 robber

  17. Tree-Depth: 2 definitions d cops 1 robber

  18. Tree-Depth: 2 definitions d cops 1 robber

  19. Tree-Depth: 2 definitions d cops 1 robber

  20. Tree-Depth: 2 definitions d cops 1 robber Cop player wins if a cop lands on the robber

  21. Tree-Depth: 2 definitions Fact: A graph has tree-depth d iff the Cop player has a winning strategy in the game using d cops

  22. Tree-Depth: 2 definitions

  23. Tree-Depth: 2 definitions

  24. Tree-Depth: 2 definitions

  25. Tree-Depth: 2 definitions Cop Wins

  26. Bounding the Number of Roots Thm [Dvorak, Giannopolou and Thilikos 12]: The class C={G:td(G)≤d} is characterized by a finite set of forbidden subgraphs, each of size at most 2^2^(d-1) Cor: Number of roots of a graph of tree-depth d is at most 2^2^(d-1)

  27. Bounding the Number of Roots H G H is forbidden subgraph for tree-depth <=d-1, and H has tree-depth d Cor: Number of roots of a graph of tree-depth d is at most 2^2^(d-1)

  28. Bounding the Number of Roots B S2 S1 … Sk

  29. Bounding the Number of Roots Si ≈Sj iff there is an isomorphism from Si U B to Sj U B which also preserves edges to B B S2 S1 … Sk

  30. Bounding the Number of Roots Thm: Deleting more than d copies of same component does not affect set of roots of the tree-depth B S2 S1 … Sk

  31. Bounding the Number of Roots Thm: Deleting more than d copies of same component does not affect set of roots of the tree-depth Idea: Never play cops in more than d copies Can “mirror” strategies using only d copies B S2 S1 … Sk

  32. Bounding the Number of Roots B WLOG G is minimal #Vertices in component containing robber (and hence #Roots) bounded by reverse induction G’ S1 S1 S1 S1 S1 S2 S2 … Sk Sk Sk

  33. Bounding the Number of Roots B WLOG G is minimal #Vertices in component containing robber (and hence #Roots) bounded by reverse induction G’ S1 S1 S1 S1 S1 S2 S2 … Sk Sk Sk

  34. Isomorphism Algorithm • Define S<T if • |S|<|T| • |S|=|T| and #s <#t • |S|=|T|, #s=#t. and • (S1…S#s)<(T1…T#t) • where S_i and T_i are inductively ordered components of S and T s

  35. Isomorphism Algorithm • Define S<T if • |S|<|T| • |S|=|T| and #s <#t • |S|=|T|, #s=#t and • (E(s,r1)..E(s,rk))< (E(t,r1)..E(t,rk)) • 4.Above equal and • (S1…S#s)<(T1…T#t) r1 s Theorem 1: Graph Isomorphism is FPT in tree-depth

  36. Extension: Subdivisions Defn: A graph has generalized tree-depth d iff it is a subdivision of a graph of tree-depth d Theorem 2: Graph Isomorphism is FPT in the generalized tree-depth

  37. Size of smallest excluded minor ? FPT ? ? Tree-Width Genus ? ? Crossing Number Path-Width Generalized Tree-Depth Tree-Depth Max Leaf Number Vertex Cover Number

  38. Questions ?

More Related