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Basis Sequence Reconfiguration in the Union of Matroids

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

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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.

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

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.

Subject Classification

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

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