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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.

References

  1. Kira Adaricheva, Chassidy Bozeman, Nancy E. Clarke, Ruth Haas, Margaret-Ellen Messinger, Karen Seyffarth, and Heather C. Smith. Reconfiguration graphs for dominating sets. In Research Trends in Graph Theory and Applications, volume 25 of Assoc. Women Math. Ser., pages 119-135. Springer, Cham, 2021. Google Scholar
  2. Akanksha Agrawal, Soumita Hait, and Amer E. Mouawad. On finding short reconfiguration sequences between independent sets. In 33rd International Symposium on Algorithms and Computation, volume 248 of LIPIcs. Leibniz Int. Proc. Inform., pages Paper No. 39, 14. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2022. Google Scholar
  3. Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, and Danny Hermelin. On problems without polynomial kernels. J. Comput. Syst. Sci., 75(8):423-434, 2009. URL: https://doi.org/10.1016/j.jcss.2009.04.001.
  4. Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernelization lower bounds by cross-composition. SIAM J. Discret. Math., 28(1):277-305, 2014. URL: https://doi.org/10.1137/120880240.
  5. Robert Connelly, Erik D. Demaine, and Günter Rote. Blowing up polygonal linkages. Discrete Comput. Geom., 30:205-239, 2003. Google Scholar
  6. Daniel W. Cranston and Reem Mahmoud. In most 6-regular toroidal graphs all 5-colorings are Kempe equivalent. European J. Combin., 104:Paper No. 103532, 21, 2022. Google Scholar
  7. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  8. Holger Dell and Dieter van Melkebeek. Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM, 61(4):23:1-23:27, 2014. URL: https://doi.org/10.1145/2629620.
  9. Quentin Deschamps, Carl Feghali, František Kardoš, Clément Legrand-Duchesne, and Théo Pierron. Strengthening a theorem of Meyniel, 2022. Google Scholar
  10. Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Monographs in Computer Science. Springer, 1999. URL: https://doi.org/10.1007/978-1-4612-0515-9.
  11. Andrew Drucker. New limits to classical and quantum instance compression. SIAM J. Comput., 44(5):1443-1479, 2015. URL: https://doi.org/10.1137/130927115.
  12. Cem Evrendilek. Vertex separators for partitioning a graph. Sensors, 8(2):635-657, 2008. Google Scholar
  13. Lester R. Ford and Delbert R. Fulkerson. Flows in Networks. Princeton University Press, USA, 2010. Google Scholar
  14. Lance Fortnow and Rahul Santhanam. Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci., 77(1):91-106, 2011. URL: https://doi.org/10.1016/j.jcss.2010.06.007.
  15. Bin Fu and Zhixiang Chen. Sublinear time width-bounded separators and their application to the protein side-chain packing problem. J. Comb. Optim., 15(4):387-407, 2008. Google Scholar
  16. Guilherme C. M. Gomes, Sérgio H. Nogueira, and Vinícius F. dos Santos. Some results on vertex separator reconfiguration, 2020. Google Scholar
  17. Takehiro Ito, Erik D. Demaine, Xiao Zhou, and Takao Nishizeki. Approximability of partitioning graphs with supply and demand. J. Discrete Algorithms, 6(4):627-650, 2008. Google Scholar
  18. Takehiro Ito, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, and Yoshio Okamoto. Shortest reconfiguration of perfect matchings via alternating cycles. SIAM J. Discrete Math., 36(2):1102-1123, 2022. Google Scholar
  19. Enver Kayaaslan, Ali Pinar, Ümit Çatalyürek, and Cevdet Aykanat. Partitioning hypergraphs in scientific computing applications through vertex separators on graphs. SIAM J. Sci. Comput., 34(2):A970-A992, 2012. Google Scholar
  20. Charles E. Leiserson. Area-efficient graph layouts. In 21st Annual Symposium on Foundations of Computer Science, pages 270-281, 1980. Google Scholar
  21. Karl Menger. Zur allgemeinen Kurventheorie. Fundamenta Mathematicae, 10:96-115, 1927. URL: http://eudml.org/doc/211191.
  22. István Miklós and Heather Smith. Sampling and counting genome rearrangement scenarios. BMC Bioinformatics, 16, 2015. Google Scholar
  23. Xie Xian-fen, Gu Wan-rong, He Yi-chen, and Mao Yi-jun. Matrix transformation and factorization based on graph partitioning by vertex separator for recommendation. Computer Science, 49(6):272-279, 2022. Google Scholar
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