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Shortest Reconfiguration of Colorings Under Kempe Changes

Authors Marthe Bonamy, Marc Heinrich, Takehiro Ito , Yusuke Kobayashi, Haruka Mizuta, Moritz Mühlenthaler, Akira Suzuki , Kunihiro Wasa

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LIPIcs.STACS.2020.35.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Combinatorial Reconfiguration
  • Graph Algorithms
  • Graph Coloring
  • Kempe Equivalence

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Abstract

A k-coloring of a graph maps each vertex of the graph to a color in {1, 2, …, k}, such that no two adjacent vertices receive the same color. Given a k-coloring of a graph, a Kempe change produces a new k-coloring by swapping the colors in a bicolored connected component. We investigate the complexity of finding the smallest number of Kempe changes needed to transform a given k-coloring into another given k-coloring. We show that this problem admits a polynomial-time dynamic programming algorithm on path graphs, which turns out to be highly non-trivial. Furthermore, the problem is NP-hard even on star graphs and we show that on such graphs it admits a constant-factor approximation algorithm and is fixed-parameter tractable when parameterized by the number k of colors. The hardness result as well as the algorithmic results are based on the notion of a canonical transformation.

Cite As Get BibTex

Marthe Bonamy, Marc Heinrich, Takehiro Ito, Yusuke Kobayashi, Haruka Mizuta, Moritz Mühlenthaler, Akira Suzuki, and Kunihiro Wasa. Shortest Reconfiguration of Colorings Under Kempe Changes. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 35:1-35:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.STACS.2020.35

Author Details

Marthe Bonamy
  • CNRS, LaBRI, Université de Bordeaux, Talence, France
Marc Heinrich
  • Université Lyon 1, LIRIS, UMR5205, France
Takehiro Ito
  • Graduate School of Information Sciences, Tohoku University, Japan
Yusuke Kobayashi
  • Research Institute for Mathematical Sciences, Kyoto University, Japan
Haruka Mizuta
  • Graduate School of Information Sciences, Tohoku University, Japan
Moritz Mühlenthaler
  • Laboratoire G-SCOP, Grenoble INP, Université Grenoble Alpes, France
Akira Suzuki
  • Graduate School of Information Sciences, Tohoku University, Japan
Kunihiro Wasa
  • National Institute of Informatics, Tokyo, Japan

Funding

  • Ito, Takehiro: Partially supported by JST CREST Grant Number JPMJCR1402, and JSPS KAKENHI Grant Numbers JP18H04091 and JP19K11814, Japan.
  • Kobayashi, Yusuke: Supported by JST ACT-I Grant Number JPMJPR17UB, and JSPS KAKENHI Grant Numbers JP16K16010 and JP18H05291, Japan.
  • Suzuki, Akira: Partially supported by JST CREST Grant Number JPMJCR1402, and JSPS KAKENHI Grant Numbers JP17K12636 and JP18H04091, Japan.
  • Wasa, Kunihiro: Partially supported by JST CREST Grant Numbers JPMJCR18K3 and JPMJCR1401, and JSPS KAKENHI Grant Number 19K20350, Japan.

Acknowledgements

This work is partially supported by JSPS and MEAE-MESRI under the Japan-France Integrated Action Program (SAKURA).

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