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Minimum Separator Reconfiguration

Authors Guilherme C. M. Gomes , Clément Legrand-Duchesne , Reem Mahmoud, Amer E. Mouawad , Yoshio Okamoto , Vinicius F. dos Santos , Tom C. van der Zanden

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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • minimum separators
  • combinatorial reconfiguration
  • parameterized complexity
  • kernelization

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Abstract

We study the problem of reconfiguring one minimum s-t-separator A into another minimum s-t-separator B in some n-vertex graph G containing two non-adjacent vertices s and t. We consider several variants of the problem as we focus on both the token sliding and token jumping models. Our first contribution is a polynomial-time algorithm that computes (if one exists) a minimum-length sequence of slides transforming A into B. We additionally establish that the existence of a sequence of jumps (which need not be of minimum length) can be decided in polynomial time (by an algorithm that also outputs a witnessing sequence when one exists). In contrast, and somewhat surprisingly, we show that deciding if a sequence of at most 𝓁 jumps can transform A into B is an NP-complete problem. To complement this negative result, we investigate the parameterized complexity of what we believe to be the two most natural parameterized counterparts of the latter problem; in particular, we study the problem of computing a minimum-length sequence of jumps when parameterized by the size k of the minimum s-t-separators and when parameterized by the number 𝓁 of jumps. For the first parameterization, we show that the problem is fixed-parameter tractable, but does not admit a polynomial kernel unless NP ⊆ coNP/poly. We complete the picture by designing a kernel with 𝒪(𝓁²) vertices and edges for the length 𝓁 of the sequence as a parameter.

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Guilherme C. M. Gomes, Clément Legrand-Duchesne, Reem Mahmoud, Amer E. Mouawad, Yoshio Okamoto, Vinicius F. dos Santos, and Tom C. van der Zanden. Minimum Separator Reconfiguration. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 9:1-9:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.IPEC.2023.9

Author Details

Guilherme C. M. Gomes
  • Department of Computer Science, Federal, University of Minas Gerais, Belo Horizonte, Brazil
Clément Legrand-Duchesne
  • LaBRI, CNRS, Université de Bordeaux, France
Reem Mahmoud
  • Virginia Commonwealth University, Richmond, VA, USA
Amer E. Mouawad
  • Department of Computer Science, American University of Beirut, Beirut, Lebanon
Yoshio Okamoto
  • Graduate School of Informatics and Engineering, The University of Electro-Communications, Chofu, Japan
Vinicius F. dos Santos
  • Department of Computer Science, Federal, University of Minas Gerais, Belo Horizonte, Brazil
Tom C. van der Zanden
  • Department of Data Analytics and Digitalisation, Maastricht University, The Netherlands

Funding

  • Okamoto, Yoshio: JSPS KAKENHI Grant Numbers JP20K11670, JP20H05795, JP23K10982.
  • F. dos Santos, Vinicius: FAPEMIG grant APQ-01707-21 and CNPq grants 406036/2021-7 and 312069/2021-9.

Acknowledgements

This work started during the Combinatorial Reconfiguration Workshop (CoRe 2022) which was hosted at the Banff International Research Station for Mathematical Innovation and Discovery (BIRS), Alberta, Canada, from May 8-13, 2022. We would like to thank everyone who made this collaboration possible and, in particular, the organizers Daniel Cranston, Marthe Bonamy, Moritz Mühlenthaler, Naomi Nishimura, Nicolas Bousquet, Ryuhei Uehara, and Takehiro Ito for their continuous support of the combinatorial reconfiguration community in general.

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