the parameterized complexity of graph cyclabilityosebje.famnit.upr.si/~martin.milanic/agtac 2015...
TRANSCRIPT
![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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/1.jpg)
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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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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/3.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/4.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/5.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/6.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/7.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/8.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/9.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/10.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/11.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/12.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/13.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/14.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/15.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/16.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/17.jpg)
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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
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/19.jpg)
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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
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/21.jpg)
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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
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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
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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
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Problem-irrelevant vertices
Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability
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Problem-irrelevant vertices
Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability
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Problem-irrelevant vertices
Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/28.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/29.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/30.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/31.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/32.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/33.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/34.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/35.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/36.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/37.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/38.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/39.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/40.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/41.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/42.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/43.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/44.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/45.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/46.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/47.jpg)
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Color-irrelevant vertex
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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/48.jpg)
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Color-irrelevant vertex
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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/49.jpg)
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Color-irrelevant vertex
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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/50.jpg)
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Color-irrelevant vertex
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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/51.jpg)
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Color-irrelevant vertex
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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/52.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/53.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/54.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/55.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/56.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/57.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/58.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/59.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/60.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/61.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/62.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/63.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/64.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/65.jpg)
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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
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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
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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
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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
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/70.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/71.jpg)
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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](https://reader033.vdocument.in/reader033/viewer/2022052803/5f257df6e3f0423f48374188/html5/thumbnails/72.jpg)
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Thank you
Spyridon Maniatis UoAThe Parameterized Complexity of Graph Cyclability