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On the Parameterized Complexity of Reconfiguration Problems

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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 \(\mathcal {Q}\) 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 \(\mathcal {Q}\). 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 \(\ell \), 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 \(k+\ell \). We also demonstrate the W[1]-hardness of reconfiguration variants of a large class of maximization problems parameterized by \(k+\ell \), and of their corresponding deletion problems parameterized by \(\ell \); 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 \(k+\ell \).

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Acknowledgments

The second author wishes to thank Marcin Kamiński for suggesting the examination of reconfiguration in the parameterized setting.

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Authors and Affiliations

  1. Department of Informatics, University of Bergen, Bergen, Norway

    Amer E. Mouawad

  2. David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada

    Naomi Nishimura

  3. The Institute of Mathematical Sciences, Chennai, India

    Venkatesh Raman

  4. Google Waterloo, 151 Charles Street West, Suite 200, Kitchener, ON, N2G 1H6, Canada

    Narges Simjour

  5. Graduate School of Information Sciences, Tohoku University, Sendai, Japan

    Akira Suzuki

  6. CREST, JST, Saitama, Japan

    Akira Suzuki

Authors
  1. Amer E. Mouawad
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  2. Naomi Nishimura
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  3. Venkatesh Raman
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  4. Narges Simjour
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  5. Akira Suzuki
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Corresponding author

Correspondence to Amer E. Mouawad.

Additional information

A preliminary version of this paper appeared in the Proceedings of the 8th International Symposium on Parameterized and Exact Computation (IPEC 2013).

Research of the first, second, and fourth authors is supported by the Natural Science and Engineering Research Council of Canada. Research of the fifth author is supported by JSPS Grant-in-Aid for Scientific Research, Grant Number 26730001.

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Mouawad, A.E., Nishimura, N., Raman, V. et al. On the Parameterized Complexity of Reconfiguration Problems. Algorithmica 78, 274–297 (2017). https://doi.org/10.1007/s00453-016-0159-2

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