On the Parameterized Complexity of Reconfiguration Problems

We present the first results on the parameterized complexity of reconfiguration problems, where a reconfiguration variant of an optimization problem Q takes as input two feasible solutions S and T and determines if there is a sequence of reconfiguration steps, i.e. a reconfiguration sequence, that c...

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Vydané v:	Algorithmica Ročník 78; číslo 1; s. 274 - 297
Hlavní autori:	Mouawad, Amer E., Nishimura, Naomi, Raman, Venkatesh, Simjour, Narges, Suzuki, Akira
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.05.2017 Springer Nature B.V
Predmet:	Algorithm Analysis and Problem Complexity Algorithms Apexes Complexity Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graph theory Mathematics of Computing Optimization Parameterization Reconfiguration Theory of Computation Vertex sets Vertex cover Hitting set Solution space Parameterized complexity Reconfiguration
ISSN:	0178-4617, 1432-0541
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Abstract	We present the first results on the parameterized complexity of reconfiguration problems, where a reconfiguration variant of an optimization problem Q takes as input two feasible solutions S and T and determines if there is a sequence of reconfiguration steps, i.e. a reconfiguration sequence, that can be applied to transform S into T such that each step results in a feasible solution to Q . For most of the results in this paper, S and T are sets of vertices of a given graph and a reconfiguration step adds or removes a vertex. Our study is motivated by results establishing that for many NP -hard problems, the classical complexity of reconfiguration is PSPACE -complete. We address the question for several important graph properties under two natural parameterizations: k , a bound on the size of solutions, and ℓ , a bound on the length of reconfiguration sequences. Our first general result is an algorithmic paradigm, the reconfiguration kernel, used to obtain fixed-parameter tractable algorithms for reconfiguration variants of Vertex Cover and, more generally, Bounded Hitting Set and Feedback Vertex Set , all parameterized by k . In contrast, we show that reconfiguring Unbounded Hitting Set is W[2] -hard when parameterized by k + ℓ . We also demonstrate the W[1] -hardness of reconfiguration variants of a large class of maximization problems parameterized by k + ℓ , and of their corresponding deletion problems parameterized by ℓ ; in doing so, we show that there exist problems in FPT when parameterized by k , but whose reconfiguration variants are W[1] -hard when parameterized by k + ℓ .
AbstractList	We present the first results on the parameterized complexity of reconfiguration problems, where a reconfiguration variant of an optimization problem Q takes as input two feasible solutions S and T and determines if there is a sequence of reconfiguration steps, i.e. a reconfiguration sequence, that can be applied to transform S into T such that each step results in a feasible solution to Q . For most of the results in this paper, S and T are sets of vertices of a given graph and a reconfiguration step adds or removes a vertex. Our study is motivated by results establishing that for many NP -hard problems, the classical complexity of reconfiguration is PSPACE -complete. We address the question for several important graph properties under two natural parameterizations: k, a bound on the size of solutions, and ℓ , a bound on the length of reconfiguration sequences. Our first general result is an algorithmic paradigm, the reconfiguration kernel, used to obtain fixed-parameter tractable algorithms for reconfiguration variants of Vertex Cover and, more generally, Bounded Hitting Set and Feedback Vertex Set , all parameterized by k. In contrast, we show that reconfiguring Unbounded Hitting Set is W[2] -hard when parameterized by k + ℓ . We also demonstrate the W[1] -hardness of reconfiguration variants of a large class of maximization problems parameterized by k + ℓ , and of their corresponding deletion problems parameterized by ℓ ; in doing so, we show that there exist problems in FPT when parameterized by k, but whose reconfiguration variants are W[1] -hard when parameterized by k + ℓ . We present the first results on the parameterized complexity of reconfiguration problems, where a reconfiguration variant of an optimization problem Q takes as input two feasible solutions S and T and determines if there is a sequence of reconfiguration steps, i.e. a reconfiguration sequence, that can be applied to transform S into T such that each step results in a feasible solution to Q . For most of the results in this paper, S and T are sets of vertices of a given graph and a reconfiguration step adds or removes a vertex. Our study is motivated by results establishing that for many NP -hard problems, the classical complexity of reconfiguration is PSPACE -complete. We address the question for several important graph properties under two natural parameterizations: k , a bound on the size of solutions, and ℓ , a bound on the length of reconfiguration sequences. Our first general result is an algorithmic paradigm, the reconfiguration kernel, used to obtain fixed-parameter tractable algorithms for reconfiguration variants of Vertex Cover and, more generally, Bounded Hitting Set and Feedback Vertex Set , all parameterized by k . In contrast, we show that reconfiguring Unbounded Hitting Set is W[2] -hard when parameterized by k + ℓ . We also demonstrate the W[1] -hardness of reconfiguration variants of a large class of maximization problems parameterized by k + ℓ , and of their corresponding deletion problems parameterized by ℓ ; in doing so, we show that there exist problems in FPT when parameterized by k , but whose reconfiguration variants are W[1] -hard when parameterized by k + ℓ .
Author	Mouawad, Amer E. Raman, Venkatesh Simjour, Narges Suzuki, Akira Nishimura, Naomi
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Snippet	We present the first results on the parameterized complexity of reconfiguration problems, where a reconfiguration variant of an optimization problem Q takes as... We present the first results on the parameterized complexity of reconfiguration problems, where a reconfiguration variant of an optimization problem Q takes as...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Apexes Complexity Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graph theory Mathematics of Computing Optimization Parameterization Reconfiguration Theory of Computation Vertex sets
Title	On the Parameterized Complexity of Reconfiguration Problems
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