Kernel lower bound for the k -domatic partition problem Rmi - PowerPoint PPT Presentation

Kernel lower bound for the k -domatic partition problem Rémi Watrigant joint work with Sylvain Guillemot and Christophe Paul LIRMM, Montpellier, France AGAPE Workshop, February 6-10, 2012, Montpellier, France Rémi Watrigant Kernel lower bound for the k -domatic partition problem 1/27

Contents Kernels, domatic partition 1 hypergraph-2-colorability 2 Transformation to k -domatic partition 3 Conclusion, open question 4 Rémi Watrigant Kernel lower bound for the k -domatic partition problem 2/27

Kernels, domatic partition Rémi Watrigant Kernel lower bound for the k -domatic partition problem 3/27

Kernels, domatic partition Kernel Given a parameterized problem Q ⊆ Σ ∗ × N , a kernel for Q is a polynomial algorithm with: input: an instance ( x , k ) of Q output: an instance ( x ′ , k ′ ) of Q such that: ( x , k ) ∈ Q ⇔ ( x ′ , k ′ ) ∈ Q | x ′ | , k ′ ≤ f ( k ) for some function f Rémi Watrigant Kernel lower bound for the k -domatic partition problem 3/27

Kernels, domatic partition Kernel Given a parameterized problem Q ⊆ Σ ∗ × N , a kernel for Q is a polynomial algorithm with: input: an instance ( x , k ) of Q output: an instance ( x ′ , k ′ ) of Q such that: ( x , k ) ∈ Q ⇔ ( x ′ , k ′ ) ∈ Q | x ′ | , k ′ ≤ f ( k ) for some function f Theorem Q ∈ FPT ⇔ Q has a kernel Rémi Watrigant Kernel lower bound for the k -domatic partition problem 3/27

k -domatic partition (for fixed k ∈ N ) Input : a graph G = ( V , E ) Question : Is there a k -partition of V : { V 1 , ..., V k } such that each V i is a dominating set of G ? Rémi Watrigant Kernel lower bound for the k -domatic partition problem 4/27

b b b b b b b b b b b k -domatic partition (for fixed k ∈ N ) Input : a graph G = ( V , E ) Question : Is there a k -partition of V : { V 1 , ..., V k } such that each V i is a dominating set of G ? k = 3 Rémi Watrigant Kernel lower bound for the k -domatic partition problem 4/27

b b b b b b b b b b b k -domatic partition (for fixed k ∈ N ) Input : a graph G = ( V , E ) Question : Is there a k -partition of V : { V 1 , ..., V k } such that each V i is a dominating set of G ? k = 3 Rémi Watrigant Kernel lower bound for the k -domatic partition problem 4/27

k -domatic partition (for fixed k ∈ N ) Input : a graph G = ( V , E ) Question : Is there a k -partition of V : { V 1 , ..., V k } such that each V i is a dominating set of G ? Known results: Rémi Watrigant Kernel lower bound for the k -domatic partition problem 4/27

k -domatic partition (for fixed k ∈ N ) Input : a graph G = ( V , E ) Question : Is there a k -partition of V : { V 1 , ..., V k } such that each V i is a dominating set of G ? Known results: Any graph admits a 1-domatic partition and a 2-domatic partition Rémi Watrigant Kernel lower bound for the k -domatic partition problem 4/27

k -domatic partition (for fixed k ∈ N ) Input : a graph G = ( V , E ) Question : Is there a k -partition of V : { V 1 , ..., V k } such that each V i is a dominating set of G ? Known results: Any graph admits a 1-domatic partition and a 2-domatic partition for any fixed k ≥ 3, the problem is NP -complete [Garey, Johnson, Tarjan, 76] ⇒ k is useless as a parameter (for FPT, kernels...) Rémi Watrigant Kernel lower bound for the k -domatic partition problem 4/27

k -domatic partition (for fixed k ∈ N ) Input : a graph G = ( V , E ) Question : Is there a k -partition of V : { V 1 , ..., V k } such that each V i is a dominating set of G ? Known results: Any graph admits a 1-domatic partition and a 2-domatic partition for any fixed k ≥ 3, the problem is NP -complete [Garey, Johnson, Tarjan, 76] ⇒ k is useless as a parameter (for FPT, kernels...) FPT when parameterized by treewidth ( G ) (MSO formula) Rémi Watrigant Kernel lower bound for the k -domatic partition problem 4/27

k -domatic partition (for fixed k ∈ N ) Input : a graph G = ( V , E ) Question : Is there a k -partition of V : { V 1 , ..., V k } such that each V i is a dominating set of G ? Known results: Any graph admits a 1-domatic partition and a 2-domatic partition for any fixed k ≥ 3, the problem is NP -complete [Garey, Johnson, Tarjan, 76] ⇒ k is useless as a parameter (for FPT, kernels...) FPT when parameterized by treewidth ( G ) (MSO formula) 3 -domatic partition does not admit a polynomial kernel when parameterized by treewidth ( G ) [Bodlaender et al. 2009] (unless all coNP problems have a distillation algorithm...) Rémi Watrigant Kernel lower bound for the k -domatic partition problem 4/27

Hierarchy of parameters Theorem [Bodlaender et al. 2009] 3 -domatic partition does not admit a polynomial kernel when parameterized by treewidth ( G ) (unless all coNP problems have a distillation algorithm...) What about larger parameters ? Rémi Watrigant Kernel lower bound for the k -domatic partition problem 5/27

Hierarchy of parameters Theorem [Bodlaender et al. 2009] 3 -domatic partition does not admit a polynomial kernel when parameterized by treewidth ( G ) (unless all coNP problems have a distillation algorithm...) What about larger parameters ? treewidth(G) Rémi Watrigant Kernel lower bound for the k -domatic partition problem 5/27

Hierarchy of parameters Theorem [Bodlaender et al. 2009] 3 -domatic partition does not admit a polynomial kernel when parameterized by treewidth ( G ) (unless all coNP problems have a distillation algorithm...) What about larger parameters ? treewidth(G) ≤ poly(VC(G)) Rémi Watrigant Kernel lower bound for the k -domatic partition problem 5/27

Hierarchy of parameters Theorem [Bodlaender et al. 2009] 3 -domatic partition does not admit a polynomial kernel when parameterized by treewidth ( G ) (unless all coNP problems have a distillation algorithm...) What about larger parameters ? treewidth(G) poly(FVS(G)) poly(VC(G)) ≤ ≤ Rémi Watrigant Kernel lower bound for the k -domatic partition problem 5/27

Hierarchy of parameters Theorem [Bodlaender et al. 2009] 3 -domatic partition does not admit a polynomial kernel when parameterized by treewidth ( G ) (unless all coNP problems have a distillation algorithm...) What about larger parameters ? treewidth(G) poly(FVS(G)) poly(VC(G)) ≤ ≤ treewidth ≤ 0 + kv Rémi Watrigant Kernel lower bound for the k -domatic partition problem 5/27

Hierarchy of parameters Theorem [Bodlaender et al. 2009] 3 -domatic partition does not admit a polynomial kernel when parameterized by treewidth ( G ) (unless all coNP problems have a distillation algorithm...) What about larger parameters ? treewidth(G) poly(FVS(G)) poly(VC(G)) ≤ ≤ treewidth ≤ 1 + kv treewidth ≤ 0 + kv Rémi Watrigant Kernel lower bound for the k -domatic partition problem 5/27

Hierarchy of parameters Theorem [Bodlaender et al. 2009] 3 -domatic partition does not admit a polynomial kernel when parameterized by treewidth ( G ) (unless all coNP problems have a distillation algorithm...) What about larger parameters ? treewidth(G) ≤ poly(FVS(G)) poly(VC(G)) ≤ ≤ treewidth ≤ t + kv treewidth ≤ 1 + kv treewidth ≤ 0 + kv Rémi Watrigant Kernel lower bound for the k -domatic partition problem 5/27

Hierarchy of parameters Theorem [Bodlaender et al. 2009] 3 -domatic partition does not admit a polynomial kernel when parameterized by treewidth ( G ) (unless all coNP problems have a distillation algorithm...) What about larger parameters ? treewidth(G) ≤ poly(FVS(G)) poly(VC(G)) ≤ ≤ treewidth ≤ t + kv treewidth ≤ 1 + kv treewidth ≤ 0 + kv Our result: For any fixed k ≥ 3, k -domatic partition does not admit a polynomial kernel when parameterized by the size of a vertex cover of G (unless coNP ⊆ NP / Poly ) Rémi Watrigant Kernel lower bound for the k -domatic partition problem 5/27

Our result: For any fixed k ≥ 3, k -domatic partition does not admit a polynomial kernel when parameterized by the size of a vertex cover of G (unless coNP ⊆ NP / Poly ) Sketch of proof: cross-composition of hypergraph-2-colorability to itself ⇒ no polynomial kernel for hypergraph-2-colorability (parameterized by the number of vertices) polynomial time and parameter transformation to k -domatic partition Rémi Watrigant Kernel lower bound for the k -domatic partition problem 6/27

Contents Kernels, domatic partition 1 hypergraph-2-colorability 2 Transformation to k -domatic partition 3 Conclusion, open question 4 Rémi Watrigant Kernel lower bound for the k -domatic partition problem 7/27

Lower bound for hypergraph-2-colorability hypergraph-2-colorability Input : a hypergraph H = ( V , E ) Question : Is there a bipartition of V into ( V 1 , V 2 ) such that each hyperedge has at least one vertex in V 1 and one vertex in V 2 ? Parameter : n = | V | Rémi Watrigant Kernel lower bound for the k -domatic partition problem 8/27

Lower bound for hypergraph-2-colorability hypergraph-2-colorability Input : a hypergraph H = ( V , E ) Question : Is there a bipartition of V into ( V 1 , V 2 ) such that each hyperedge has at least one vertex in V 1 and one vertex in V 2 ? Parameter : n = | V | Theorem [Bodlaender, Jansen, Kratsch, 2011] If there exists a cross-composition from an NP -complete problem A to a parameterized problem Q , then Q does not admit a polynomial kernel unless coNP ⊆ NP / Poly Rémi Watrigant Kernel lower bound for the k -domatic partition problem 8/27