A single-exponential FPT algorithm for the K4-minor cover problem

Given a graph G and a parameter k∈N, the parameterized K4-minor cover problem asks whether at most k vertices can be deleted to turn G into a K4-minor-free graph, or equivalently in a graph of treewidth at most 2. This problem is inspired by two well-studied parameterized vertex deletion problems, V...

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Published in:Journal of computer and system sciences Vol. 81; no. 1; pp. 186 - 207
Main Authors: Kim, Eun Jung, Paul, Christophe, Philip, Geevarghese
Format: Journal Article
Language:English
Published: Elsevier Inc 01.02.2015
Elsevier
Subjects:
[formula omitted]-minor cover
Computer Science
Discrete Mathematics
Fixed-parameter tractable algorithms
Treewidth-2 Deletion
Treewidth-2 Deletion
Fixed-parameter tractable algorithms
K4-minor cover
FPT time
Vertex Cover
ISSN:0022-0000, 1090-2724
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Abstract Given a graph G and a parameter k∈N, the parameterized K4-minor cover problem asks whether at most k vertices can be deleted to turn G into a K4-minor-free graph, or equivalently in a graph of treewidth at most 2. This problem is inspired by two well-studied parameterized vertex deletion problems, Vertex Cover and Feedback Vertex Set, which can also be expressed as Treewidth-tVertex Deletion problems: t=0 for Vertex Cover and t=1 for Feedback Vertex Set. While single-exponential FPT algorithms, i.e. running in 2O(k)⋅nO(1) time, are known for these two latter problems, it was open whether the K4-minor cover problem could be solved in single-exponential FPT time. This paper answers this question in the affirmative. Observe that it is known to be unlikely that Treewidth-tVertex Deletion can be solved in time 2o(k)⋅nO(1). •We provide an efficient FPT algorithm for the K4-minor cover problem.•It combines iterative compression with protrusion reduction and branching.•It extends previous algorithms for Vertex Cover and Feedback Vertex Set.
AbstractList Given a graph G and a parameter k∈N, the parameterized K4-minor cover problem asks whether at most k vertices can be deleted to turn G into a K4-minor-free graph, or equivalently in a graph of treewidth at most 2. This problem is inspired by two well-studied parameterized vertex deletion problems, Vertex Cover and Feedback Vertex Set, which can also be expressed as Treewidth-tVertex Deletion problems: t=0 for Vertex Cover and t=1 for Feedback Vertex Set. While single-exponential FPT algorithms, i.e. running in 2O(k)⋅nO(1) time, are known for these two latter problems, it was open whether the K4-minor cover problem could be solved in single-exponential FPT time. This paper answers this question in the affirmative. Observe that it is known to be unlikely that Treewidth-tVertex Deletion can be solved in time 2o(k)⋅nO(1). •We provide an efficient FPT algorithm for the K4-minor cover problem.•It combines iterative compression with protrusion reduction and branching.•It extends previous algorithms for Vertex Cover and Feedback Vertex Set.
Given an input graph G and an integer k, the parameterized K4 -minor cover problem asks whether there is a set S of at most k vertices whose deletion results in a K4 -minor-free graph, or equivalently in a graph of treewidth at most 2. This problem is inspired by two well-studied parameterized vertex deletion problems, Vertex Cover and Feedback Vertex Set , which can also be expressed as Treewidth- t Vertex Deletion problems: t = 0 for Vertex Cover and t = 1 for Feedback Vertex Set . While a single-exponential FPT algorithm has been known for a long time for Vertex Cover , such an algorithm for Feedback Vertex Set was devised comparatively recently. While it is known to be unlikely that Treewidth- t Vertex Deletion can be solved in time co ( k ) nO (1) , it was open whether the K4 -minor cover could be solved in single-exponential FPT time, i.e. in ck nO (1) time.
Author Paul, Christophe
Kim, Eun Jung
Philip, Geevarghese
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Issue 1
Keywords Treewidth-2 Deletion
Fixed-parameter tractable algorithms
K4-minor cover
FPT time
Vertex Cover
Language English
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Snippet Given a graph G and a parameter k∈N, the parameterized K4-minor cover problem asks whether at most k vertices can be deleted to turn G into a K4-minor-free...
Given an input graph G and an integer k, the parameterized K4 -minor cover problem asks whether there is a set S of at most k vertices whose deletion results...
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StartPage 186
SubjectTerms [formula omitted]-minor cover
Computer Science
Discrete Mathematics
Fixed-parameter tractable algorithms
Treewidth-2 Deletion
Title A single-exponential FPT algorithm for the K4-minor cover problem
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