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. . The Parameterized Complexity of Graph Cyclability Spyridon Maniatis AGTAC, Koper, June 2015 Joint work with Petr A. Golovach, Marcin Kamiński, and Dimitrios M. Thilikos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Page 1: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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......The Parameterized Complexity of Graph Cyclability

Spyridon Maniatis

AGTAC, Koper, June 2015

Joint work with Petr A. Golovach, Marcin Kamiński, and Dimitrios M. Thilikos

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Page 2: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Introduction

We study (from the algorithmic point of view) a connectivityrelated parameter, namely cyclability [V. Chvátal, 1973].Can be thought as of a quantitive measure of Hamiltonicity(or a way to unify connectivity and Hamiltonicity):

.Cyclability..

......

A graph G is k -cyclable if every k vertices of V(G) lie in acommon cycle. The cyclability of G is the maximum integer k forwhich G is k -cyclable.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 3: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Introduction

We study (from the algorithmic point of view) a connectivityrelated parameter, namely cyclability [V. Chvátal, 1973].Can be thought as of a quantitive measure of Hamiltonicity(or a way to unify connectivity and Hamiltonicity):

.Cyclability..

......

A graph G is k -cyclable if every k vertices of V(G) lie in acommon cycle. The cyclability of G is the maximum integer k forwhich G is k -cyclable.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 4: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Introduction

We study (from the algorithmic point of view) a connectivityrelated parameter, namely cyclability [V. Chvátal, 1973].Can be thought as of a quantitive measure of Hamiltonicity(or a way to unify connectivity and Hamiltonicity):

.Cyclability..

......

A graph G is k -cyclable if every k vertices of V(G) lie in acommon cycle. The cyclability of G is the maximum integer k forwhich G is k -cyclable.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 5: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Introduction

Natural question: Is there an efficient (polynomial?)algorithm computing the cyclability of a graph?

NO, because hamiltonian cycle is NP-hard (even forcubic planar graphs).

From the parameterized complexity point of view?

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 6: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Introduction

Natural question: Is there an efficient (polynomial?)algorithm computing the cyclability of a graph?

NO, because hamiltonian cycle is NP-hard (even forcubic planar graphs).

From the parameterized complexity point of view?

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 7: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Introduction

Natural question: Is there an efficient (polynomial?)algorithm computing the cyclability of a graph?

NO, because hamiltonian cycle is NP-hard (even forcubic planar graphs).

From the parameterized complexity point of view?

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 8: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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The Parameterized Problem

.p-Cyclability...

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Input: A graph G and a positive integer k .Parameter: k.Question: Is G k -cyclable?

We actually consider, for technical reasons, the more general,annotated version of the problem:

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 9: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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The Annotated Version

.p-Annotated Cyclability...

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Input: A graph G, a set R ⊆ V(G) and a positive integer k .Parameter: k.Question: Is it true that for every S ⊆ R with |S| ≤ k, there existsa cycle of G that meets all the vertices of S?

Of course, when R = V(G) we have an instance of the initialproblem.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 10: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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The Annotated Version

.p-Annotated Cyclability...

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Input: A graph G, a set R ⊆ V(G) and a positive integer k .Parameter: k.Question: Is it true that for every S ⊆ R with |S| ≤ k, there existsa cycle of G that meets all the vertices of S?

Of course, when R = V(G) we have an instance of the initialproblem.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 11: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Our results

We study the parameterized complexity of Cyclability. Ourresults are:

1 Cyclability is co-W[1]-hard (even for split-graphs), whenparameterized by k .

2 The problem is in FPT when restricted to the class of planargraphs.

3 No polynomial kernel unless NP ⊆ co-NP/poly, whenrestricted to cubic planar graphs.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 12: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Our results

We study the parameterized complexity of Cyclability. Ourresults are:

1 Cyclability is co-W[1]-hard (even for split-graphs), whenparameterized by k .

2 The problem is in FPT when restricted to the class of planargraphs.

3 No polynomial kernel unless NP ⊆ co-NP/poly, whenrestricted to cubic planar graphs.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 13: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Our results

We study the parameterized complexity of Cyclability. Ourresults are:

1 Cyclability is co-W[1]-hard (even for split-graphs), whenparameterized by k .

2 The problem is in FPT when restricted to the class of planargraphs.

3 No polynomial kernel unless NP ⊆ co-NP/poly, whenrestricted to cubic planar graphs.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 14: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Our results

We study the parameterized complexity of Cyclability. Ourresults are:

1 Cyclability is co-W[1]-hard (even for split-graphs), whenparameterized by k .

2 The problem is in FPT when restricted to the class of planargraphs.

3 No polynomial kernel unless NP ⊆ co-NP/poly, whenrestricted to cubic planar graphs.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 15: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Theorem 1

.Theorem 1..

......The p-Cyclability problem is co-W[1]-hard. This also holds ifthe inputs are restricted to be split graphs.

Reduction of the k -Clique problem to:.p-Cyclability complement...

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Input: A split graph G and a positive integer k .Parameter: k.Question: Is there an S ⊆ V(G), |S| ≤ k s.t. there is no cycle ofG that contains all vertices of S?

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 16: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Theorem 1

.Theorem 1..

......The p-Cyclability problem is co-W[1]-hard. This also holds ifthe inputs are restricted to be split graphs.

Reduction of the k -Clique problem to:.p-Cyclability complement...

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Input: A split graph G and a positive integer k .Parameter: k.Question: Is there an S ⊆ V(G), |S| ≤ k s.t. there is no cycle ofG that contains all vertices of S?

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 17: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Theorem 1

.Theorem 1..

......The p-Cyclability problem is co-W[1]-hard. This also holds ifthe inputs are restricted to be split graphs.

Reduction of the k -Clique problem to:.p-Cyclability complement...

......

Input: A split graph G and a positive integer k .Parameter: k.Question: Is there an S ⊆ V(G), |S| ≤ k s.t. there is no cycle ofG that contains all vertices of S?

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 18: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Theorem 2

.Theorem 2..

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The p-Cyclability problem, when parameterized by k , is inFPT when its inputs are restricted to be planar graphs. Moreover,the corresponding FPT-algorithm runs in 22

O(k2 log k) · n2 steps.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 19: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Irrelevant vertex technique

We refer to p-Annotated Cyclability, restricted toplanar graphs, as problem Π.Main idea of our algorithm: Application of the irrelevantvertex technique (introduced by Robertson, Seymour, GMXXII, 2012).

For our purposes, we actually consider two kinds of irrelevantvertex.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 20: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Irrelevant vertex technique

We refer to p-Annotated Cyclability, restricted toplanar graphs, as problem Π.Main idea of our algorithm: Application of the irrelevantvertex technique (introduced by Robertson, Seymour, GMXXII, 2012).

For our purposes, we actually consider two kinds of irrelevantvertex.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 21: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Irrelevant vertices

.Problem-irrelevant vertex..

......Let (G,R, k) be an instance for Π. Then vertex v ∈ V(G) is calledproblem-irrelevant for Π, if (G,R, k) ∈ Π ⇔ (G \ v,R, k) ∈ Π .

.Color-irrelevant vertex..

......Let (G,R, k) be an instance for Π. Then vertex v ∈ R is calledcolor-irrelevant for Π, if (G,R, k) ∈ Π ⇔ (G,R \ v, k) ∈ Π.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 22: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Irrelevant vertices

.Problem-irrelevant vertex..

......Let (G,R, k) be an instance for Π. Then vertex v ∈ V(G) is calledproblem-irrelevant for Π, if (G,R, k) ∈ Π ⇔ (G \ v,R, k) ∈ Π .

.Color-irrelevant vertex..

......Let (G,R, k) be an instance for Π. Then vertex v ∈ R is calledcolor-irrelevant for Π, if (G,R, k) ∈ Π ⇔ (G,R \ v, k) ∈ Π.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 23: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Irrelevant vertices

.Problem-irrelevant vertex..

......Let (G,R, k) be an instance for Π. Then vertex v ∈ V(G) is calledproblem-irrelevant for Π, if (G,R, k) ∈ Π ⇔ (G \ v,R, k) ∈ Π .

.Color-irrelevant vertex..

......Let (G,R, k) be an instance for Π. Then vertex v ∈ R is calledcolor-irrelevant for Π, if (G,R, k) ∈ Π ⇔ (G,R \ v, k) ∈ Π.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 24: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Problem-irrelevant vertices

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 25: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Problem-irrelevant vertices

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 26: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Problem-irrelevant vertices

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 27: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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The Algorithm (First step)

Check if tw(G) is upper bounded by an (appropriate) functionof k . If YES, solve using dynamic programming.

Else, we show that there exists a cycle of the plane embeddingthat contains a “large” subdivided wall H as a subgraph andthe part of G that is surrounded by the perimeter of H hasbounded treewidth.

Find in H a sequence C of “many” concentric cycles that areall traversed by “many” disjoint paths of H. We call such astructure a railed annulus.

‘

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 28: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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The Algorithm (First step)

Check if tw(G) is upper bounded by an (appropriate) functionof k . If YES, solve using dynamic programming.

Else, we show that there exists a cycle of the plane embeddingthat contains a “large” subdivided wall H as a subgraph andthe part of G that is surrounded by the perimeter of H hasbounded treewidth.

Find in H a sequence C of “many” concentric cycles that areall traversed by “many” disjoint paths of H. We call such astructure a railed annulus.

‘

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 29: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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The Algorithm (First step)

Check if tw(G) is upper bounded by an (appropriate) functionof k . If YES, solve using dynamic programming.

Else, we show that there exists a cycle of the plane embeddingthat contains a “large” subdivided wall H as a subgraph andthe part of G that is surrounded by the perimeter of H hasbounded treewidth.

Find in H a sequence C of “many” concentric cycles that areall traversed by “many” disjoint paths of H. We call such astructure a railed annulus.

‘

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 30: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Railed annulus

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 31: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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The Algorithm (Second step)

Check whether in the railed annulus there exists a “large”(“bidimensional”) part (function of k2) not containing anycolored vertices.

If YES, pick a problem-irrelevant vertex (we prove that itexists) and produce a smaller equivalent instance.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 32: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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The Algorithm (Second step)

Check whether in the railed annulus there exists a “large”(“bidimensional”) part (function of k2) not containing anycolored vertices.

If YES, pick a problem-irrelevant vertex (we prove that itexists) and produce a smaller equivalent instance.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 33: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Big uncolored part

irrelevant

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 34: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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The Algorithm (Third step)

Else, we know that the annotated vertices are “uniformly”distributed in the railed annulus.

There exists an annotated vertex w ∈ R in the “centre” of theannulus.

We set up a sequence of instances of Π “around” w, eachcorresponding to the graph “cropped” by the interior of somecycles of C.

We show that in each of them there exists a “sufficientlylarge” (function of k) railed annulus.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 35: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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The Algorithm (Third step)

Else, we know that the annotated vertices are “uniformly”distributed in the railed annulus.

There exists an annotated vertex w ∈ R in the “centre” of theannulus.

We set up a sequence of instances of Π “around” w, eachcorresponding to the graph “cropped” by the interior of somecycles of C.

We show that in each of them there exists a “sufficientlylarge” (function of k) railed annulus.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 36: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Sequence of instances

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 37: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Sequence of instances

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 38: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Sequence of instances

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 39: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Sequence of instances

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 40: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Sequence of instances

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 41: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Sequence of instances

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 42: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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The Algorithm (Fourth step)

Obtain an answer for every instance, produced in the secondstep, by a sequence of dynamic programming calls.

If there exists a no-instance report that the initial instance isa no-instance and stop.

Otherwise we prove that the annotated “central” vertex thatwe fixed earlier is color irrelevant.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 43: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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The Algorithm (Fourth step)

Obtain an answer for every instance, produced in the secondstep, by a sequence of dynamic programming calls.

If there exists a no-instance report that the initial instance isa no-instance and stop.

Otherwise we prove that the annotated “central” vertex thatwe fixed earlier is color irrelevant.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 44: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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The Algorithm (Fourth step)

Obtain an answer for every instance, produced in the secondstep, by a sequence of dynamic programming calls.

If there exists a no-instance report that the initial instance isa no-instance and stop.

Otherwise we prove that the annotated “central” vertex thatwe fixed earlier is color irrelevant.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 45: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Color-irrelevant vertex

(G,R, k) ∈ Π ⇒ (G,R \ w, k) ∈ Π : Trivial.

(G,R \ w, k) ∈ Π ⇒ (G,R, k) ∈ Π ?

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 46: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Color-irrelevant vertex

(G,R, k) ∈ Π ⇒ (G,R \ w, k) ∈ Π : Trivial.

(G,R \ w, k) ∈ Π ⇒ (G,R, k) ∈ Π ?

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 47: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 48: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 49: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 50: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 51: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 52: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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To sum up

After linear number of executions of the procedure:Input rejected orTreewidth is small → Dynamic programming

Something of the above will occur after O(n) steps because ateach iteration we reject the input, we “lose” a vertex or we uncolora vertex.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 53: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Irrelevant vertices for the PDPP [Adler et al.]

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 54: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Main combinatorial statement

The following theorem enables us to find problem-irrelevantvertices:.Theorem..

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Let G be a graph embedded on the sphere S0, that is the union ofr ≥ 2 concentric cycles C = {C1, . . . ,Cr} and one more cycle C ofG. Assume that C is tight in G, T ∩ V(Cr) = ∅ and the cycliclinkage L = (C,T) is strongly vital in G. Then r ≤ 16 · |T| − 1.

Intuition: If there exists a cycle that meets S ⊆ R, then there alsoexists one that meets S and does not “go deep” in a bidimensionalgraph that does not contain any vertices of S.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 55: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Main combinatorial statement

The following theorem enables us to find problem-irrelevantvertices:.Theorem..

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Let G be a graph embedded on the sphere S0, that is the union ofr ≥ 2 concentric cycles C = {C1, . . . ,Cr} and one more cycle C ofG. Assume that C is tight in G, T ∩ V(Cr) = ∅ and the cycliclinkage L = (C,T) is strongly vital in G. Then r ≤ 16 · |T| − 1.

Intuition: If there exists a cycle that meets S ⊆ R, then there alsoexists one that meets S and does not “go deep” in a bidimensionalgraph that does not contain any vertices of S.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 56: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Main combinatorial statement

The following theorem enables us to find problem-irrelevantvertices:.Theorem..

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Let G be a graph embedded on the sphere S0, that is the union ofr ≥ 2 concentric cycles C = {C1, . . . ,Cr} and one more cycle C ofG. Assume that C is tight in G, T ∩ V(Cr) = ∅ and the cycliclinkage L = (C,T) is strongly vital in G. Then r ≤ 16 · |T| − 1.

Intuition: If there exists a cycle that meets S ⊆ R, then there alsoexists one that meets S and does not “go deep” in a bidimensionalgraph that does not contain any vertices of S.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 57: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Dynamic Programming

Some more about the DP for Cyclability:

Non-trivial DP algorithm (22tw·log tw).

Causes the double exponential dependance on k2 log k.

DP improvement → Overall improvement of the algorithm.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 58: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Dynamic Programming

Some more about the DP for Cyclability:

Non-trivial DP algorithm (22tw·log tw).

Causes the double exponential dependance on k2 log k.

DP improvement → Overall improvement of the algorithm.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 59: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Dynamic Programming

Some more about the DP for Cyclability:

Non-trivial DP algorithm (22tw·log tw).

Causes the double exponential dependance on k2 log k.

DP improvement → Overall improvement of the algorithm.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 60: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Kernelization lower bound

Our results suggest that Cyclability, parameterized by k, isunlikely to admit a polynomial kernel, when restricted to planargraphs:.Theorem 3..

......Cyclability, parameterized by k, has no polynomial kernelunless NP ⊆ co-NP/poly, when restricted to cubic planar graphs.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 61: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Kernelization lower bound

Our results suggest that Cyclability, parameterized by k, isunlikely to admit a polynomial kernel, when restricted to planargraphs:.Theorem 3..

......Cyclability, parameterized by k, has no polynomial kernelunless NP ⊆ co-NP/poly, when restricted to cubic planar graphs.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 62: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Kernelization lower bound

Let L ⊆ Σ∗ × N be a parameterized problem..Kernelization for L..

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A kernelization for problem L is an algorithm that takes as aninstance (x, k) of L and maps it, in polynomial time, to an instance(x′, k′) such that

1 (x, k) ∈ L iff (x′, k′) ∈ L2 |x′| ≤ f(k)3 |k′| ≤ g(k)

where f and g are computable functions. Function f is the size ofthe kernel and a kernel is polynomial if the corresponding functionf is polynomial.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 63: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Kernel

(G, k)(G′, k′)

|G′| ≤ f(k)

k′ ≤ g(k)

poly(|G|, k)

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 64: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Kernelization lower boundThe proof uses the cross-composition technique (introduced byBodlaender, Jansen and Kratsch):.AND-cross-composition..

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An AND-cross-composition of L ⊆ Σ∗ into Q ∈ Σ∗ × N (w.r.t. apolynomial equivalence relation R), is an algorithm that, given tinstances x1, . . . , xt ∈ Σ∗ of L belonging to the same equivalenceclass of R, takes polynomial time in

∑ti=1 |xi| and outputs an

instance (y, k) ∈ Σ∗ × N such that:the parameter value k is polynomially bounded inmax{|x1|, . . . , |xt|}+ log t(y, k) is a yes-instance for Q iff each instance xi is ayes-instance for L for i ∈ {1, . . . , t}

We say that L AND-cross-composes into Q if a cross-compositionalgorithm exists for a suitable relation R.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 65: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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AND-cross-composition

G1 G2Gt

(G, k)

k ≤ poly(max{|x1|, . . . |xt|} + logt)

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 66: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Kernelization lower bound.Theorem..

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Assume that an NP-hard language L AND-cross-composes to aparameterized language Q. Then Q does not admit a polynomialkernel, unless NP ⊆ co-NP/poly.

.Hamiltonicity with a Given Edge..

......Input: A graph G and e ∈ E(G).Question: Does G have a hamiltonian cycle C s.t. e ∈ E(C)?

Hamiltonicity with a Given Edge is NP-complete forcubic planar graphs.Hamiltonicity with a Given Edge AND-cross-composesinto p-Cyclability.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 67: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Kernelization lower bound.Theorem..

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Assume that an NP-hard language L AND-cross-composes to aparameterized language Q. Then Q does not admit a polynomialkernel, unless NP ⊆ co-NP/poly.

.Hamiltonicity with a Given Edge..

......Input: A graph G and e ∈ E(G).Question: Does G have a hamiltonian cycle C s.t. e ∈ E(C)?

Hamiltonicity with a Given Edge is NP-complete forcubic planar graphs.Hamiltonicity with a Given Edge AND-cross-composesinto p-Cyclability.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 68: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Further research

1 Improve (if possible) the DP algorithm for Cyclability.

2 Prove completeness of Cyclability for some level of thepolynomial hierrarchy.

3 Prove completeness of p-Cyclability for some level of theW-hierrarchy.

4 Apply the irrelevant vertex technique to more(connectivity-related) problems.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 69: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Further research

1 Improve (if possible) the DP algorithm for Cyclability.

2 Prove completeness of Cyclability for some level of thepolynomial hierrarchy.

3 Prove completeness of p-Cyclability for some level of theW-hierrarchy.

4 Apply the irrelevant vertex technique to more(connectivity-related) problems.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 70: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Further research

1 Improve (if possible) the DP algorithm for Cyclability.

2 Prove completeness of Cyclability for some level of thepolynomial hierrarchy.

3 Prove completeness of p-Cyclability for some level of theW-hierrarchy.

4 Apply the irrelevant vertex technique to more(connectivity-related) problems.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 71: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Further research

1 Improve (if possible) the DP algorithm for Cyclability.

2 Prove completeness of Cyclability for some level of thepolynomial hierrarchy.

3 Prove completeness of p-Cyclability for some level of theW-hierrarchy.

4 Apply the irrelevant vertex technique to more(connectivity-related) problems.

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability

Page 72: The Parameterized Complexity of Graph Cyclabilityosebje.famnit.upr.si/~martin.milanic/AGTAC 2015 slides/Maniatis.pdf · The Parameterized Complexity of Graph Cyclability Spyridon

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Thank you

Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability


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Parameterized Algorithmics: A Graph-Theoretic Approachfernau/papers/habil.pdf · Parameterized Algorithmics: A Graph-Theoretic Approach ... 2.2 Is parameterized algorithmics the solution

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Parameterized Complexity of DPLL Search Procedures · 2020-03-30 · Parameterized Complexity of DPLL Search Procedures Olaf Beyersdor School of Computing, University of Leeds, UK

Parameterized Complexity SAT · The Gentle Revolution of Parameterized Complexity • Bioinformatics, Operations Research, Optimization, Automated Reasoning, etc." • 4 Monographs"-[Downey&Fellows

Parameterized complexity of DPLL search procedures · Parameterized complexity is a rich and informative theory, and we suggest the monographs [27,31,35] for further reading about

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Machine-based Methods in Parameterized Complexity Theorybasics.sjtu.edu.cn/~chen/papers/machine.pdf · Machine-based Methods in Parameterized Complexity Theory ... The theory is built

The Parameterized Complexity of Cascading Portfolio Schedulingpapers.nips.cc/paper/8983-the-parameterized... · Parameterized Complexity. In parameterized algorithmics [6, 4, 3, 9]

On the Space and Circuit Complexity of Parameterized …in classical computational complexity theory carried over to the world of pa-rameterized complexity theory: Parameterized complexity

The Parameterized Complexity of the Induced Matching Problem … · 2010-12-09 · The Parameterized Complexity of the Induced Matching Problem2,3 Hannes Moser1,∗ Institut fu¨r

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Parameterized Complexity and Graph Isomorphismad260/bouland.pdf · complexity classes which form the parameterized analogues of NP. 1.3 Work completed in this dissertation Graph Isomorphism

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Parameterized Complexity of Finding Small Degree ...amini/Publications/ASS.pdf · Concerning the parameterized complexity of nding regular subgraphs, Moser and Thilikos proved that

Parameterized Complexity of Neighborhood Problems in Graphs …vraman/pub/dsswat.pdf · 2007-03-26 · parameterized complexity classes on general graphs, become fixed pa-rameter