Document Open Access Logo

Reconfiguration of Labeled Matchings in Triangular Grid Graphs

Authors Naonori Kakimura , Yuta Mishima

PDF

File

Thumbnail PDF
PDF
LIPIcs.ISAAC.2024.43.pdf
  • Filesize: 0.81 MB
  • 16 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • combinatorial reconfiguration
  • matching
  • factor-critical graphs
  • sliding-block puzzles

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

This paper introduces a new reconfiguration problem of matchings in a triangular grid graph. In this problem, we are given a nearly perfect matching in which each matching edge is labeled, and aim to transform it to a target matching by sliding edges one by one. This problem is motivated to investigate the solvability of a sliding-block puzzle called "Gourds" on a hexagonal grid board, introduced by Hamersma et al. [ISAAC 2020]. The main contribution of this paper is to prove that, if a triangular grid graph is factor-critical and has a vertex of degree 6, then any two matchings can be reconfigured to each other. Moreover, for a triangular grid graph (which may not have a degree-6 vertex), we present another sufficient condition using the local connectivity. Both of our results provide broad sufficient conditions for the solvability of the Gourds puzzle on a hexagonal grid board with holes, where Hamersma et al. left it as an open question.

Cite As Get BibTex

Naonori Kakimura and Yuta Mishima. Reconfiguration of Labeled Matchings in Triangular Grid Graphs. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.43

Author Details

Naonori Kakimura
  • Department of Mathematics, Keio University, Yokohama, Japan
Yuta Mishima
  • The Japan Research Institute, Limited, Tokyo, Japan

Funding

  • Kakimura, Naonori: Supported by JSPS KAKENHI Grant Numbers JP22H05001, JP20H05795, and 23K21646, Japan and JST ERATO Grant Number JPMJER2301, Japan.

Acknowledgements

The authors are grateful to Yushi Uno for valuable discussions. We are also grateful for the anonymous referees of ISAAC 2024 for their helpful comments.

References

  1. Oswin Aichholzer, Erik D. Demaine, Matias Korman, Anna Lubiw, Jayson Lynch, Zuzana Masárová, Mikhail Rudoy, Virginia Vassilevska Williams, and Nicole Wein. Hardness of token swapping on trees. In Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, and Grzegorz Herman, editors, 30th Annual European Symposium on Algorithms, ESA 2022, September 5-9, 2022, Berlin/Potsdam, Germany, volume 244 of LIPIcs, pages 3:1-3:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.ESA.2022.3.
  2. Valentin Bartier, Nicolas Bousquet, and Amer E. Mouawad. Galactic token sliding. J. Comput. Syst. Sci., 136:220-248, 2023. URL: https://doi.org/10.1016/J.JCSS.2023.03.008.
  3. Marthe Bonamy, Nicolas Bousquet, Marc Heinrich, Takehiro Ito, Yusuke Kobayashi, Arnaud Mary, Moritz Mühlenthaler, and Kunihiro Wasa. The perfect matching reconfiguration problem. In Peter Rossmanith, Pinar Heggernes, and Joost-Pieter Katoen, editors, 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, August 26-30, 2019, Aachen, Germany, volume 138 of LIPIcs, pages 80:1-80:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPICS.MFCS.2019.80.
  4. Édouard Bonnet, Tillmann Miltzow, and Pawel Rzazewski. Complexity of token swapping and its variants. Algorithmica, 80(9):2656-2682, 2018. URL: https://doi.org/10.1007/S00453-017-0387-0.
  5. Nicolas Bousquet, Amer E. Mouawad, Naomi Nishimura, and Sebastian Siebertz. A survey on the parameterized complexity of the independent set and (connected) dominating set reconfiguration problems. CoRR, abs/2204.10526, 2022. URL: https://doi.org/10.48550/arXiv.2204.10526.
  6. Josh Brunner, Lily Chung, Erik D. Demaine, Dylan H. Hendrickson, Adam Hesterberg, Adam Suhl, and Avi Zeff. 1 X 1 rush hour with fixed blocks is pspace-complete. In Martin Farach-Colton, Giuseppe Prencipe, and Ryuhei Uehara, editors, 10th International Conference on Fun with Algorithms, FUN 2021, May 30 to June 1, 2021, Favignana Island, Sicily, Italy, volume 157 of LIPIcs, pages 7:1-7:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.FUN.2021.7.
  7. Kevin Buchin and Maike Buchin. Rolling block mazes are pspace-complete. J. Inf. Process., 20(3):719-722, 2012. URL: https://doi.org/10.2197/IPSJJIP.20.719.
  8. Jean Cardinal and Raphael Steiner. Inapproximability of shortest paths on perfect matching polytopes. In Alberto Del Pia and Volker Kaibel, editors, Integer Programming and Combinatorial Optimization - 24th International Conference, IPCO 2023, Madison, WI, USA, June 21-23, 2023, Proceedings, volume 13904 of Lecture Notes in Computer Science, pages 72-86. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-32726-1_6.
  9. Erik D. Demaine, Martin L. Demaine, Eli Fox-Epstein, Duc A. Hoang, Takehiro Ito, Hirotaka Ono, Yota Otachi, Ryuhei Uehara, and Takeshi Yamada. Linear-time algorithm for sliding tokens on trees. Theor. Comput. Sci., 600:132-142, 2015. URL: https://doi.org/10.1016/J.TCS.2015.07.037.
  10. Erik D. Demaine and Mikhail Rudoy. A simple proof that the (n² -1)-puzzle is hard. Theor. Comput. Sci., 732:80-84, 2018. URL: https://doi.org/10.1016/J.TCS.2018.04.031.
  11. Gary William Flake and Eric B. Baum. Rush hour is pspace-complete, or "why you should generously tip parking lot attendants". Theor. Comput. Sci., 270(1-2):895-911, 2002. URL: https://doi.org/10.1016/S0304-3975(01)00173-6.
  12. Valery S. Gordon, Yury L. Orlovich, and Frank Werner. Hamiltonian properties of triangular grid graphs. Discret. Math., 308(24):6166-6188, 2008. URL: https://doi.org/10.1016/J.DISC.2007.11.040.
  13. Joep Hamersma, Marc J. van Kreveld, Yushi Uno, and Tom C. van der Zanden. Gourds: A sliding-block puzzle with turning. In Yixin Cao, Siu-Wing Cheng, and Minming Li, editors, 31st International Symposium on Algorithms and Computation, ISAAC 2020, December 14-18, 2020, Hong Kong, China (Virtual Conference), volume 181 of LIPIcs, pages 33:1-33:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPICS.ISAAC.2020.33.
  14. Robert A. Hearn and Erik D. Demaine. Pspace-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theor. Comput. Sci., 343(1-2):72-96, 2005. URL: https://doi.org/10.1016/J.TCS.2005.05.008.
  15. Robert A. Hearn and Erik D. Demaine. Games, puzzles and computation. A K Peters, 2009. Google Scholar
  16. Takehiro Ito, Erik D. Demaine, Nicholas J. A. Harvey, Christos H. Papadimitriou, Martha Sideri, Ryuhei Uehara, and Yushi Uno. On the complexity of reconfiguration problems. Theor. Comput. Sci., 412(12-14):1054-1065, 2011. URL: https://doi.org/10.1016/J.TCS.2010.12.005.
  17. Takehiro Ito, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, and Yoshio Okamoto. Shortest reconfiguration of perfect matchings via alternating cycles. SIAM J. Discret. Math., 36(2):1102-1123, 2022. URL: https://doi.org/10.1137/20M1364370.
  18. Wm. Woolsey Johnson and William E. Story. Notes on the "15" puzzle. American Journal of Mathematics, 2(4):397-404, 1879. URL: http://www.jstor.org/stable/2369492.
  19. Naonori Kakimura and Yuta Mishima. Reconfiguration of labeled matchings in triangular grid graphs, 2024. URL: https://arxiv.org/abs/2409.11723.
  20. Dohan Kim. Sorting on graphs by adjacent swaps using permutation groups. Comput. Sci. Rev., 22:89-105, 2016. URL: https://doi.org/10.1016/J.COSREV.2016.09.003.
  21. L. Lovász and M.D. Plummer. Matching Theory. AMS Chelsea Publishing Series. AMS Chelsea Pub., 2009. Google Scholar
  22. Tillmann Miltzow, Lothar Narins, Yoshio Okamoto, Günter Rote, Antonis Thomas, and Takeaki Uno. Approximation and hardness of token swapping. In Piotr Sankowski and Christos D. Zaroliagis, editors, 24th Annual European Symposium on Algorithms, ESA 2016, August 22-24, 2016, Aarhus, Denmark, volume 57 of LIPIcs, pages 66:1-66:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPICS.ESA.2016.66.
  23. Naomi Nishimura. Introduction to reconfiguration. Algorithms, 11(4):52, 2018. URL: https://doi.org/10.3390/a11040052.
  24. Daniel Ratner and Manfred K. Warmuth. Finding a shortest solution for the N times N extension of the 15-puzzle is intractable. In Tom Kehler, editor, Proceedings of the 5th National Conference on Artificial Intelligence. Philadelphia, PA, USA, August 11-15, 1986. Volume 1: Science, pages 168-172. Morgan Kaufmann, 1986. URL: http://www.aaai.org/Library/AAAI/1986/aaai86-027.php.
  25. Laura Sanità. The diameter of the fractional matching polytope and its hardness implications. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 910-921, 2018. URL: https://doi.org/10.1109/FOCS.2018.00090.
  26. Alexander Schrijver. Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin, 2003. Google Scholar
  27. J. Slocum and D. Sonneveld. The 15 Puzzle Book: How It Drove the World Crazy. Slocum Puzzle Foundation, 2006. Google Scholar
  28. Jan van den Heuvel. The complexity of change. In Simon R. Blackburn, Stefanie Gerke, and Mark Wildon, editors, Surveys in Combinatorics 2013, volume 409 of London Mathematical Society Lecture Note Series, pages 127-160. Cambridge University Press, 2013. URL: https://doi.org/10.1017/CBO9781139506748.005.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact