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

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

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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

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Contents

1

Kernels, domatic partition

2

hypergraph-2-colorability

3

Transformation to k-domatic partition

4

Conclusion, open question

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 2/27

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Kernels, domatic partition

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

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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

  • utput: 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

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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

  • utput: 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

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k-domatic partition (for fixed k ∈ N)

Input : a graph G = (V , E) Question : Is there a k-partition of V : {V1, ..., Vk} such that each Vi is a dominating set of G ?

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 4/27

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k-domatic partition (for fixed k ∈ N)

Input : a graph G = (V , E) Question : Is there a k-partition of V : {V1, ..., Vk} such that each Vi is a dominating set of G ? k = 3 b b b b b b b b b

b b

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 4/27

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k-domatic partition (for fixed k ∈ N)

Input : a graph G = (V , E) Question : Is there a k-partition of V : {V1, ..., Vk} such that each Vi is a dominating set of G ? k = 3

b b

b b b b b b b b b

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 4/27

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k-domatic partition (for fixed k ∈ N)

Input : a graph G = (V , E) Question : Is there a k-partition of V : {V1, ..., Vk} such that each Vi is a dominating set of G ? Known results:

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 4/27

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k-domatic partition (for fixed k ∈ N)

Input : a graph G = (V , E) Question : Is there a k-partition of V : {V1, ..., Vk} such that each Vi 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

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k-domatic partition (for fixed k ∈ N)

Input : a graph G = (V , E) Question : Is there a k-partition of V : {V1, ..., Vk} such that each Vi 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

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k-domatic partition (for fixed k ∈ N)

Input : a graph G = (V , E) Question : Is there a k-partition of V : {V1, ..., Vk} such that each Vi 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

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k-domatic partition (for fixed k ∈ N)

Input : a graph G = (V , E) Question : Is there a k-partition of V : {V1, ..., Vk} such that each Vi 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

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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

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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

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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

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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)) poly(FVS(G))

≤ ≤

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 5/27

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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)) poly(FVS(G)) treewidth ≤ 0 + kv

≤ ≤

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 5/27

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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)) poly(FVS(G)) treewidth ≤ 0 + kv treewidth ≤ 1 + kv

≤ ≤

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 5/27

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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)) poly(FVS(G)) treewidth ≤ 0 + kv treewidth ≤ 1 + kv treewidth ≤ t + kv

≤ ≤ ≤

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 5/27

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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)) poly(FVS(G)) treewidth ≤ 0 + kv treewidth ≤ 1 + kv treewidth ≤ t + 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

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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

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Contents

1

Kernels, domatic partition

2

hypergraph-2-colorability

3

Transformation to k-domatic partition

4

Conclusion, open question

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 7/27

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Lower bound for hypergraph-2-colorability

hypergraph-2-colorability

Input : a hypergraph H = (V , E) Question : Is there a bipartition of V into (V1, V2) such that each hyperedge has at least one vertex in V1 and one vertex in V2 ? Parameter : n = |V |

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 8/27

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Lower bound for hypergraph-2-colorability

hypergraph-2-colorability

Input : a hypergraph H = (V , E) Question : Is there a bipartition of V into (V1, V2) such that each hyperedge has at least one vertex in V1 and one vertex in V2 ? 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

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Lower bound for hypergraph-2-colorability

Definition : cross-composition [Bodlaender, Jansen, Kratsch, 2011]

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 9/27

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Lower bound for hypergraph-2-colorability

Definition : cross-composition [Bodlaender, Jansen, Kratsch, 2011]

A cross-composition from a problem A ⊆ Σ∗ to a parameterized problem Q ⊆ Σ∗ × N is a polynomial algorithm with: input: a sequence of equivalent instances of A: {x1, ..., xt}

  • utput : an instance of Q: (x∗, k∗)

such that:

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 9/27

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Lower bound for hypergraph-2-colorability

Definition : cross-composition [Bodlaender, Jansen, Kratsch, 2011]

A cross-composition from a problem A ⊆ Σ∗ to a parameterized problem Q ⊆ Σ∗ × N is a polynomial algorithm with: input: a sequence of equivalent instances of A: {x1, ..., xt}

  • utput : an instance of Q: (x∗, k∗)

such that: x∗ is a positive instance of Q ⇔ ∃i ∈ {1, ..., t} such that xi is a positive instance of A k∗ ≤ poly( max

i=1...t|xi| + log t)

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 9/27

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Lower bound for hypergraph-2-colorability

Definition : cross-composition [Bodlaender, Jansen, Kratsch, 2011]

A cross-composition from a problem A ⊆ Σ∗ to a parameterized problem Q ⊆ Σ∗ × N is a polynomial algorithm with: input: a sequence of equivalent instances of A: {x1, ..., xt}

  • utput : an instance of Q: (x∗, k∗)

such that: x∗ is a positive instance of Q ⇔ ∃i ∈ {1, ..., t} such that xi is a positive instance of A k∗ ≤ poly( max

i=1...t|xi| + log t)

Equivalence relation: computable in polynomial time partition a set S into less than max

x∈S |x|O(1) classes

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 9/27

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Lower bound for hypergraph-2-colorability

Let (H1, ..., Ht) be a sequence of instances of hypergraph-2-colorability

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 10/27

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Lower bound for hypergraph-2-colorability

Let (H1, ..., Ht) be a sequence of instances of hypergraph-2-colorability Equivalence relation: |Vi| = n for all i = 1...t

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 10/27

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Lower bound for hypergraph-2-colorability

Let (H1, ..., Ht) be a sequence of instances of hypergraph-2-colorability Equivalence relation: |Vi| = n for all i = 1...t Suppose that t = 2p (p = log2 t)

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 10/27

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Lower bound for hypergraph-2-colorability

Let (H1, ..., Ht) be a sequence of instances of hypergraph-2-colorability Equivalence relation: |Vi| = n for all i = 1...t Suppose that t = 2p (p = log2 t) ⇒ we are given a sequence of 2p sets of hyperedges over V = {v1, ..., vn}

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 10/27

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Lower bound for hypergraph-2-colorability

Let (H1, ..., Ht) be a sequence of instances of hypergraph-2-colorability Equivalence relation: |Vi| = n for all i = 1...t Suppose that t = 2p (p = log2 t) ⇒ we are given a sequence of 2p sets of hyperedges over V = {v1, ..., vn} v1 v2 v3 v4 vn a1 b1 a2 b2 ap+1 bp+1

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 10/27

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Lower bound for hypergraph-2-colorability

Let (H1, ..., Ht) be a sequence of instances of hypergraph-2-colorability Equivalence relation: |Vi| = n for all i = 1...t Suppose that t = 2p (p = log2 t) ⇒ we are given a sequence of 2p sets of hyperedges over V = {v1, ..., vn} v1 v2 v3 v4 vn a1 b1 a2 b2 ap+1 bp+1

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 10/27

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Lower bound for hypergraph-2-colorability

Let (H1, ..., Ht) be a sequence of instances of hypergraph-2-colorability Equivalence relation: |Vi| = n for all i = 1...t Suppose that t = 2p (p = log2 t) ⇒ we are given a sequence of 2p sets of hyperedges over V = {v1, ..., vn} v1 v2 v3 v4 vn a1 b1 a2 b2 ap+1 bp+1 hyperedge e of H1 ⇒ 2 hyperedges in H∗

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 10/27

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Lower bound for hypergraph-2-colorability

Let (H1, ..., Ht) be a sequence of instances of hypergraph-2-colorability Equivalence relation: |Vi| = n for all i = 1...t Suppose that t = 2p (p = log2 t) ⇒ we are given a sequence of 2p sets of hyperedges over V = {v1, ..., vn} v1 v2 v3 v4 vn a1 b1 a2 b2 ap+1 bp+1 hyperedge e of H1 ⇒ 2 hyperedges in H∗ binary representation of 1

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 10/27

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Lower bound for hypergraph-2-colorability

Let (H1, ..., Ht) be a sequence of instances of hypergraph-2-colorability Equivalence relation: |Vi| = n for all i = 1...t Suppose that t = 2p (p = log2 t) ⇒ we are given a sequence of 2p sets of hyperedges over V = {v1, ..., vn} v1 v2 v3 v4 vn a1 b1 a2 b2 ap+1 bp+1 hyperedge e of Hj ⇒ 2 hyperedges in H∗ binary representation of j

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 10/27

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Lower bound for hypergraph-2-colorability

Let (H1, ..., Ht) be a sequence of instances of hypergraph-2-colorability Equivalence relation: |Vi| = n for all i = 1...t Suppose that t = 2p (p = log2 t) ⇒ we are given a sequence of 2p sets of hyperedges over V = {v1, ..., vn} v1 v2 v3 v4 vn a1 b1 a2 b2 ap+1 bp+1 hyperedge ei of Hj ⇒ 2 hyperedges in H∗ binary representation of j

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 10/27

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Lower bound for hypergraph-2-colorability

v1 v2 v3 v4 vn a1 b1 a2 b2 ap+1 bp+1 hyperedge ei of Hj ⇒ 2 hyperedges in H∗ binary representation of j

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 11/27

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Lower bound for hypergraph-2-colorability

v1 v2 v3 v4 vn a1 b1 a2 b2 ap+1 bp+1 hyperedge ei of Hj ⇒ 2 hyperedges in H∗ binary representation of j Suppose Hj is a positive instance: there exists a 2-coloring that covers all hyperedges of Hj

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 11/27

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Lower bound for hypergraph-2-colorability

v1 v2 v3 v4 vn a1 b1 a2 b2 ap+1 bp+1 hyperedge ei of Hj ⇒ 2 hyperedges in H∗ binary representation of j

1 1 2 2 1

Suppose Hj is a positive instance: there exists a 2-coloring that covers all hyperedges of Hj

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 12/27

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Lower bound for hypergraph-2-colorability

v1 v2 v3 v4 vn a1 b1 a2 b2 ap+1 bp+1 hyperedge ei of Hj ⇒ 2 hyperedges in H∗ binary representation of j

1 1 2 2 1 1 1 2 2 1 2

Suppose Hj is a positive instance: there exists a 2-coloring that covers all hyperedges of Hj

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 13/27

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Lower bound for hypergraph-2-colorability

v1 v2 v3 v4 vn a1 b1 a2 b2 ap+1 bp+1 Suppose H∗ is a positive instance

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 14/27

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Lower bound for hypergraph-2-colorability

v1 v2 v3 v4 vn a1 b1 a2 b2 ap+1 bp+1

1 2 2 1 1 2 1 1 2 2 1

Suppose H∗ is a positive instance

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 15/27

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Lower bound for hypergraph-2-colorability

v1 v2 v3 v4 vn a1 b1 a2 b2 ap+1 bp+1 hyperedge ei of Hj ⇒ 2 hyperedges in H∗ binary representation of j

1 1 2 2 1 1 1 2 2 1 2

Suppose H∗ is a positive instance

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 16/27

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Lower bound for hypergraph-2-colorability

v1 v2 v3 v4 vn a1 b1 a2 b2 ap+1 bp+1 hyperedge ei of Hj ⇒ 2 hyperedges in H∗ binary representation of j Finally : the number of vertices (parameter) is polynomial in the size of the biggest instance of the sequence + log t

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 17/27

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Contents

1

Kernels, domatic partition

2

hypergraph-2-colorability

3

Transformation to k-domatic partition

4

Conclusion, open question

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 18/27

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Transformation to k-domatic partition

(proof for k = 3, but can be extended for every fixed k ≥ 3) Let H = (V , E) be an hypergraph, with V = {v1, ..., vn} and E = {e1, ..., em} We build the following graph:

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 19/27

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Transformation to k-domatic partition

(proof for k = 3, but can be extended for every fixed k ≥ 3) Let H = (V , E) be an hypergraph, with V = {v1, ..., vn} and E = {e1, ..., em} We build the following graph: e1 ej em v1 vi v ′

1

v ′

i

v ′

n

p clique vn

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 19/27

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Transformation to k-domatic partition

G ′ has a 3-domatic partition ⇔ H has a proper 2-coloring. e1 ej em v1 vi v ′

1

v ′

i

v ′

n

p clique vn

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 20/27

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Transformation to k-domatic partition

G ′ has a 3-domatic partition ⇔ H has a proper 2-coloring. e1 ej em v1 vi v ′

1

v ′

i

v ′

n

p clique vn

1

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 21/27

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Transformation to k-domatic partition

G ′ has a 3-domatic partition ⇔ H has a proper 2-coloring. e1 ej em v1 vi v ′

1

v ′

i

v ′

n

p clique vn

1 2 or 3

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 22/27

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Transformation to k-domatic partition

G ′ has a 3-domatic partition ⇔ H has a proper 2-coloring. e1 ej em v1 vi v ′

1

v ′

i

v ′

n

p clique vn

1 2 or 3 1

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 23/27

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Transformation to k-domatic partition

Finally : the clique is a vertex cover (parameter) of size n + 1 e1 ej em v1 vi v ′

1

v ′

i

v ′

n

p clique vn

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 24/27

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Contents

1

Kernels, domatic partition

2

hypergraph-2-colorability

3

Transformation to k-domatic partition

4

Conclusion, open question

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 25/27

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Conclusion, open questions

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 26/27

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Conclusion, open questions

Future work using "hierarchies of parameters": not only negative results ! vertex cover

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 26/27

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Conclusion, open questions

Future work using "hierarchies of parameters": not only negative results ! vertex cover

◮ no poly kernel when parameterized by Treewidth ◮ cubic kernel when parameterized by FeedbackVertexSet (Treewidth ≤ 1 + kv) Rémi Watrigant Kernel lower bound for the k-domatic partition problem 26/27

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Conclusion, open questions

Future work using "hierarchies of parameters": not only negative results ! vertex cover

◮ no poly kernel when parameterized by Treewidth ◮ cubic kernel when parameterized by FeedbackVertexSet (Treewidth ≤ 1 + kv)

⇒ open for Treewidth ≤ t + kv (for t ≥ 2)

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 26/27

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Conclusion, open questions

Future work using "hierarchies of parameters": not only negative results ! vertex cover

◮ no poly kernel when parameterized by Treewidth ◮ cubic kernel when parameterized by FeedbackVertexSet (Treewidth ≤ 1 + kv)

⇒ open for Treewidth ≤ t + kv (for t ≥ 2)

considering other hierarchies :

Rémi Watrigant Kernel lower bound for the k-domatic partition problem 26/27

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Conclusion, open questions

Future work using "hierarchies of parameters": not only negative results ! vertex cover

◮ no poly kernel when parameterized by Treewidth ◮ cubic kernel when parameterized by FeedbackVertexSet (Treewidth ≤ 1 + kv)

⇒ open for Treewidth ≤ t + kv (for t ≥ 2)

considering other hierarchies :

◮ distance to other invariants (CliqueWidth, ∗ − width) Rémi Watrigant Kernel lower bound for the k-domatic partition problem 26/27

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Conclusion, open questions

Future work using "hierarchies of parameters": not only negative results ! vertex cover

◮ no poly kernel when parameterized by Treewidth ◮ cubic kernel when parameterized by FeedbackVertexSet (Treewidth ≤ 1 + kv)

⇒ open for Treewidth ≤ t + kv (for t ≥ 2)

considering other hierarchies :

◮ distance to other invariants (CliqueWidth, ∗ − width) ◮ here, distance = set of vertices to remove ⋆ set of edges to remove ⋆ set of edges to edit... Rémi Watrigant Kernel lower bound for the k-domatic partition problem 26/27

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Thank you for your attention!

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