Document Open Access Logo

Basis Sequence Reconfiguration in the Union of Matroids

Authors Tesshu Hanaka , Yuni Iwamasa , Yasuaki Kobayashi , Yuto Okada , Rin Saito

PDF

File

Thumbnail PDF
PDF
LIPIcs.ISAAC.2024.38.pdf
  • Filesize: 1.08 MB
  • 16 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • Combinatorial reconfiguration
  • Matroids
  • Polynomial-time algorithm
  • Inapproximability

Metrics

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

Abstract

Given a graph G and two spanning trees T and T' in G, normalSpanning Tree Reconfiguration asks whether there is a step-by-step transformation from T to T' such that all intermediates are also spanning trees of G, by exchanging an edge in T with an edge outside T at a single step. This problem is naturally related to matroid theory, which shows that there always exists such a transformation for any pair of T and T'. Motivated by this example, we study the problem of transforming a sequence of spanning trees into another sequence of spanning trees. We formulate this problem in the language of matroid theory: Given two sequences of bases of matroids, the goal is to decide whether there is a transformation between these sequences. We design a polynomial-time algorithm for this problem, even if the matroids are given as basis oracles. To complement this algorithmic result, we show that the problem of finding a shortest transformation is NP-hard to approximate within a factor of c log n for some constant c > 0, where n is the total size of the ground sets of the input matroids.

Cite As Get BibTex

Tesshu Hanaka, Yuni Iwamasa, Yasuaki Kobayashi, Yuto Okada, and Rin Saito. Basis Sequence Reconfiguration in the Union of Matroids. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.38

Author Details

Tesshu Hanaka
  • Faculty of Information Science and Electrical Engineering, Kyushu University, Fukuoka, Japan
Yuni Iwamasa
  • Graduate School of Informatics, Kyoto University, Japan
Yasuaki Kobayashi
  • Faculty of Information Science and Technology, Hokkaido University, Sapporo, Japan
Yuto Okada
  • Graduate School of Informatics, Nagoya University, Japan
Rin Saito
  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan

Funding

  • Hanaka, Tesshu: Supported by JSPS KAKENHI Grant Numbers JP21K17707, JP21H05852, JP22H00513, and JP23H04388.
  • Iwamasa, Yuni: Supported by JSPS KAKENHI Grant Numbers JP22K17854, JP24K02901, JP24K21315.
  • Kobayashi, Yasuaki: Supported by JSPS KAKENHI Grant Numbers JP20H00595, JP23K28034, JP24H00686, and JP24H00697.
  • Okada, Yuto: Supported by JST SPRING Grant Number JPMJSP2125.
  • Saito, Rin: Supported by JST SPRING Grant Number JPMJSP2114.

References

  1. Kristóf Bérczi, Bence Mátravölgyi, and Tamás Schwarcz. Reconfiguration of basis pairs in regular matroids. In Bojan Mohar, Igor Shinkar, and Ryan O'Donnell, editors, Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, Vancouver, BC, Canada, June 24-28, 2024, pages 1653-1664. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649660.
  2. Jonah Blasiak. The toric ideal of a graphic matroid is generated by quadrics. Comb., 28(3):283-297, 2008. URL: https://doi.org/10.1007/S00493-008-2256-6.
  3. Marthe Bonamy and Nicolas Bousquet. Recoloring graphs via tree decompositions. Eur. J. Comb., 69:200-213, 2018. URL: https://doi.org/10.1016/J.EJC.2017.10.010.
  4. Paul S. Bonsma and Luis Cereceda. Finding paths between graph colourings: Pspace-completeness and superpolynomial distances. Theor. Comput. Sci., 410(50):5215-5226, 2009. URL: https://doi.org/10.1016/J.TCS.2009.08.023.
  5. Nicolas Bousquet, Felix Hommelsheim, Yusuke Kobayashi, Moritz Mühlenthaler, and Akira Suzuki. Feedback vertex set reconfiguration in planar graphs. Theor. Comput. Sci., 979:114188, 2023. URL: https://doi.org/10.1016/J.TCS.2023.114188.
  6. Martin Farber, B. Richter, and H. Shank. Edge-disjoint spanning trees: A connectedness theorem. J. Graph Theory, 9(3):319-324, 1985. URL: https://doi.org/10.1002/JGT.3190090303.
  7. Tatsuhiko Hatanaka, Takehiro Ito, and Xiao Zhou. The coloring reconfiguration problem on specific graph classes. IEICE Trans. Inf. Syst., 102-D(3):423-429, 2019. URL: https://doi.org/10.1587/TRANSINF.2018FCP0005.
  8. Jan van den Heuvel. The complexity of change. In 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.
  9. 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.
  10. Takehiro Ito, Yuni Iwamasa, Naonori Kakimura, Yusuke Kobayashi, Shun-ichi Maezawa, Yuta Nozaki, Yoshio Okamoto, and Kenta Ozeki. Rerouting planar curves and disjoint paths. In Proceedings of the 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023), volume 261 of LIPIcs, pages 81:1-81:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ICALP.2023.81.
  11. Takehiro Ito, Yuni Iwamasa, Naoyuki Kamiyama, Yasuaki Kobayashi, Yusuke Kobayashi, Shun-ichi Maezawa, and Akira Suzuki. Reconfiguration of time-respecting arborescences. In Proceedings of the 18th Algorithms and Data Structures Symposium (WADS 2023), volume 14079 of Lecture Notes in Computer Science, pages 521-532. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-38906-1_34.
  12. Takehiro Ito, Yuni Iwamasa, Yasuaki Kobayashi, Yu Nakahata, Yota Otachi, and Kunihiro Wasa. Reconfiguring (non-spanning) arborescences. Theor. Comput. Sci., 943:131-141, 2023. URL: https://doi.org/10.1016/J.TCS.2022.12.007.
  13. Yusuke Kobayashi, Ryoga Mahara, and Tamás Schwarcz. Reconfiguration of the union of arborescences. In Proceedings of the 34th International Symposium on Algorithms and Computation, (ISAAC 2023), volume 283 of LIPIcs, pages 48:1-48:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ISAAC.2023.48.
  14. Kazuo Murota. Matrices and Matroids for Systems Analysis. Springer, Berlin, Heidelberg, 2010. URL: https://doi.org/10.1007/978-3-642-03994-2.
  15. Naomi Nishimura. Introduction to reconfiguration. Algorithms, 11(4):52, 2018. URL: https://doi.org/10.3390/A11040052.
  16. James G. Oxley. Matroid Theory (Oxford Graduate Texts in Mathematics). Oxford University Press, Inc., USA, 2006. Google Scholar
  17. Ran Raz and Shmuel Safra. A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In Proceedings of the 29th Annual ACM Symposium on the Theory of Computing (STOC 1997), pages 475-484. ACM, 1997. URL: https://doi.org/10.1145/258533.258641.
  18. Rin Saito, Hiroshi Eto, Takehiro Ito, and Ryuhei Uehara. Reconfiguration of vertex-disjoint shortest paths on graphs. In Proceedings of the 17th International Conference and Workshops on Algorithms and Computation (WALCOM 2023), volume 13973 of Lecture Notes in Computer Science, pages 191-201. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-27051-2_17.
  19. Alexander Schrijver. Combinatorial Optimization: Polyhedra and Efficiency. Springer, Heidelberg, 2003. Google Scholar
  20. Neil L. White. A unique exchange property for bases. Linear Algebra and its Applications, 31:81-91, 1980. URL: https://doi.org/10.1016/0024-3795(80)90209-8.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

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