Csr2011 june16 16_30_golovach
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  • 1. Introduction Classification of computational complexity Open problems Computing vertex-surjective homomorphisms to partially reflexive trees Petr A. Golovach, Dani¨l Paulusma and Jian Song e School of Engineering and Computing Sciences, Durham University The 6th International Computer Science Symposium in Russia June 14 – 18, 2011, St. Petersburg, Russia
  • 2. Introduction Classification of computational complexity Open problemsOutline 1 Introduction Basic definitions Our results 2 Classification of computational complexity Polynomial cases Hardness 3 Open problems
  • 3. Introduction Classification of computational complexity Open problemsHomomorphisms Definition (Homomorphisms) A homomorphism from a graph G to a graph H is a mapping f : VG → VH that maps adjacent vertices of G to adjacent vertices of H, i.e., f (u)f (v ) ∈ EH whenever uv ∈ EG .
  • 4. Introduction Classification of computational complexity Open problemsHomomorphisms Definition (Homomorphisms) A homomorphism from a graph G to a graph H is a mapping f : VG → VH that maps adjacent vertices of G to adjacent vertices of H, i.e., f (u)f (v ) ∈ EH whenever uv ∈ EG . Problem (H-Coloring) The problem H-Coloring tests whether a given graph G allows a homomorphism to a graph H called the target.
  • 5. Introduction Classification of computational complexity Open problemsHomomorphisms Definition (Homomorphisms) A homomorphism from a graph G to a graph H is a mapping f : VG → VH that maps adjacent vertices of G to adjacent vertices of H, i.e., f (u)f (v ) ∈ EH whenever uv ∈ EG . Problem (H-Coloring) The problem H-Coloring tests whether a given graph G allows a homomorphism to a graph H called the target. Theorem (Hell-Neˇetˇil dichotomy theorem, 1990) s r The H-Coloring problem is solvable in polynomial time if H is bipartite, and NP-complete otherwise.
  • 6. Introduction Classification of computational complexity Open problemsSurjective homomorphisms Definition (Surjective homomorphisms) A homomorphism f from a irreflexive graph G to a partially reflexive graph H is surjective if for each x ∈ VH there exists at least one vertex u ∈ VG with f (u) = x.
  • 7. Introduction Classification of computational complexity Open problemsSurjective homomorphisms Definition (Surjective homomorphisms) A homomorphism f from a irreflexive graph G to a partially reflexive graph H is surjective if for each x ∈ VH there exists at least one vertex u ∈ VG with f (u) = x. Definition (Partially reflexive graphs) We say that a vertex incident to a self-loop is reflexive, whereas vertices with no self-loop are said to be irreflexive. A graph that contains zero or more reflexive vertices is called partially reflexive. In particular, a graph is reflexive if all its vertices are reflexive, and a graph is irreflexive if all its vertices are irreflexive.
  • 8. Introduction Classification of computational complexity Open problemsSurjective homomorphisms Definition (Surjective homomorphisms) A homomorphism f from a irreflexive graph G to a partially reflexive graph H is surjective if for each x ∈ VH there exists at least one vertex u ∈ VG with f (u) = x. Definition (Partially reflexive graphs) We say that a vertex incident to a self-loop is reflexive, whereas vertices with no self-loop are said to be irreflexive. A graph that contains zero or more reflexive vertices is called partially reflexive. In particular, a graph is reflexive if all its vertices are reflexive, and a graph is irreflexive if all its vertices are irreflexive. Problem (Surjective H-Coloring) The problem Surjective H-Coloring tests whether a given graph G allows a surjective homomorphism to a graph H.
  • 9. Introduction Classification of computational complexity Open problemsRelated problems A homomorphism f from a graph G to a graph H is locally surjective if f becomes surjective when restricted to the neighborhood of every vertex u of G .
  • 10. Introduction Classification of computational complexity Open problemsRelated problems A homomorphism f from a graph G to a graph H is locally surjective if f becomes surjective when restricted to the neighborhood of every vertex u of G . Let G and H be two graphs with a list L(u) ⊆ VH associated to each vertex u ∈ VG . Then a homomorphism f from G to H is a list-homomorphism with respect to the lists L if f (u) ∈ L(u) for all u ∈ VG .
  • 11. Introduction Classification of computational complexity Open problemsRelated problems A homomorphism f from a graph G to a graph H is locally surjective if f becomes surjective when restricted to the neighborhood of every vertex u of G . Let G and H be two graphs with a list L(u) ⊆ VH associated to each vertex u ∈ VG . Then a homomorphism f from G to H is a list-homomorphism with respect to the lists L if f (u) ∈ L(u) for all u ∈ VG . Let H be an induced subgraph of a graph G . A homomorphism f from a graph G to H is a retraction from G to H if f (h) = h for all h ∈ VH .
  • 12. Introduction Classification of computational complexity Open problemsRelated problems A homomorphism f from a graph G to a graph H is locally surjective if f becomes surjective when restricted to the neighborhood of every vertex u of G . Let G and H be two graphs with a list L(u) ⊆ VH associated to each vertex u ∈ VG . Then a homomorphism f from G to H is a list-homomorphism with respect to the lists L if f (u) ∈ L(u) for all u ∈ VG . Let H be an induced subgraph of a graph G . A homomorphism f from a graph G to H is a retraction from G to H if f (h) = h for all h ∈ VH . A homomorphism from a graph G to a graph H is called edge-surjective or a compaction if for any edge xy ∈ EH with x = y there exists an edge uv ∈ EG with f (u) = x and f (v ) = y .
  • 13. Introduction Classification of computational complexity Open problemsProblems that can be expressed via surjectivehomomorphisms Stable Cutset H G
  • 14. Introduction Classification of computational complexity Open problemsProblems that can be expressed via surjectivehomomorphisms Independent Set 1 k H G
  • 15. Introduction Classification of computational complexity Open problemsOur results We give a complete classification of the computational complexity of the Surjective H-Coloring problem when H is a partially reflexive tree.
  • 16. Introduction Classification of computational complexity Open problemsOur results We give a complete classification of the computational complexity of the Surjective H-Coloring problem when H is a partially reflexive tree. We analyze the running time of the polynomial-time solvable cases and for connected n-vertex graphs, we find a running time of nk+O(1) , where k is the number of leaves of H, and observe that this running time is asymptotically optimal.
  • 17. Introduction Classification of computational complexity Open problemsOur results We give a complete classification of the computational complexity of the Surjective H-Coloring problem when H is a partially reflexive tree. We analyze the running time of the polynomial-time solvable cases and for connected n-vertex graphs, we find a running time of nk+O(1) , where k is the number of leaves of H, and observe that this running time is asymptotically optimal. But we prove that for these cases, Surjective H-Coloring parameterized by |VH | is FPT on graphs of bounded expansion.
  • 18. Introduction Classification of computational complexity Open problemsClassification of computational complexity Theorem For any fixed partially reflexive tree H, the Surjective H-Coloring problem is polynomial time solvable if the set of reflexive vertices induces a connected subgraph (we call such graphs loop-connected), and NP-complete otherwise.
  • 19. Introduction Classification of computational complexity Open problemsClassification of computational complexity Polynomial and NP-complete cases Polynomial NP-complete
  • 20. Introduction Classification of computational complexity Open problemsPolynomial cases Theorem Let H be a loop-connected tree with k leaves. For n-vertex connected graphs, Surjective H-Coloring can be solved in time nk+O(1) .
  • 21. Introduction Classification of computational complexity Open problemsSketch of the proof Special homomorphisms u1 u5 x y z r u2 H u3 u4
  • 22. Introduction Classification of computational complexity Open problemsSketch of the proof Special homomorphisms u1 u5 x y z r u2 H u3 u4 u1 u5 u4 u2 u3
  • 23. Introduction Classification of computational complexity Open problemsSketch of the proof Special homomorphisms u1 u5 x y z r u2 H u3 u4 u1 u5 u4 u2 u3
  • 24. Introduction Classification of computational complexity Open problemsSketch of the proof Special homomorphisms u1 u5 x y z r u2 H u3 u4 u1 u5 u4 u2 u3
  • 25. Introduction Classification of computational complexity Open problemsSketch of the proof Special homomorphisms u1 u5 x y z r u2 H u3 u4 u1 u5 u4 u2 u3
  • 26. Introduction Classification of computational complexity Open problemsParameterized complexity Theorem Let H be a loop-connected tree with k leaves. For n-vertex connected graphs, Surjective H-Coloring can be solved in time nk+O(1) , i.e. this problem is in XP when parameterized by the number of leaves.
  • 27. Introduction Classification of computational complexity Open problemsParameterized complexity Independent Set 1 k Sk G
  • 28. Introduction Classification of computational complexity Open problemsParameterized complexity Proposition Surjective Sk -Coloring is W[1]-complete when parameterized by k.
  • 29. Introduction Classification of computational complexity Open problemsParameterized complexity Proposition Surjective Sk -Coloring is W[1]-complete when parameterized by k. Proposition Surjective Sk -Coloring cannot be solved in f (k) · no(k) time on n-vertex graphs, unless the Exponential Time Hypothesis collapses.
  • 30. Introduction Classification of computational complexity Open problemsParameterized complexity Theorem Let H be a loop-connected tree. Then Surjective H-Coloring is FPT for any graphs of bounded expansion when parameterized by |VH |.
  • 31. Introduction Classification of computational complexity Open problemsIdea of the proof Special homomorphisms u1 u5 x y z r u2 H u3 u4 u1 u5 u4 u2 u3
  • 32. Introduction Classification of computational complexity Open problemsNP-complete cases Theorem For any fixed partially reflexive tree H that is not loop-connected, the Surjective H-Coloring problem is NP-complete.
  • 33. Introduction Classification of computational complexity Open problemsIdea of the proof Stable Cutset H G
  • 34. Introduction Classification of computational complexity Open problemsOpen problems Give a complete complexity classification of the Surjective H-Coloring.
  • 35. Introduction Classification of computational complexity Open problemsOpen problems Give a complete complexity classification of the Surjective H-Coloring. What can be said about special case when H is a partially reflexive cycle?
  • 36. Introduction Classification of computational complexity Open problemsOpen problems Give a complete complexity classification of the Surjective H-Coloring. What can be said about special case when H is a partially reflexive cycle? Are H-Compaction, H-Retraction and Surjective H-Coloring polynomially equivalent to each other for each target graph H?
  • 37. Introduction Classification of computational complexity Open problems Thank You!