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Rerouting Planar Curves and Disjoint Paths

Authors Takehiro Ito , Yuni Iwamasa , Naonori Kakimura , Yusuke Kobayashi , Shun-ichi Maezawa , Yuta Nozaki , Yoshio Okamoto , Kenta Ozeki

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Disjoint paths
  • combinatorial reconfiguration
  • planar graphs

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Abstract

In this paper, we consider a transformation of k disjoint paths in a graph. For a graph and a pair of k disjoint paths 𝒫 and 𝒬 connecting the same set of terminal pairs, we aim to determine whether 𝒫 can be transformed to 𝒬 by repeatedly replacing one path with another path so that the intermediates are also k disjoint paths. The problem is called Disjoint Paths Reconfiguration. We first show that Disjoint Paths Reconfiguration is PSPACE-complete even when k = 2. On the other hand, we prove that, when the graph is embedded on a plane and all paths in 𝒫 and 𝒬 connect the boundaries of two faces, Disjoint Paths Reconfiguration can be solved in polynomial time. The algorithm is based on a topological characterization for rerouting curves on a plane using the algebraic intersection number. We also consider a transformation of disjoint s-t paths as a variant. We show that the disjoint s-t paths reconfiguration problem in planar graphs can be determined in polynomial time, while the problem is PSPACE-complete in general.

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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 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 81:1-81:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.81

Author Details

Takehiro Ito
  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan
Yuni Iwamasa
  • Graduate School of Informatics, Kyoto University, Japan
Naonori Kakimura
  • Faculty of Science and Technology, Keio University, Yokohama, Japan
Yusuke Kobayashi
  • Research Institute for Mathematical Sciences, Kyoto University, Japan
Shun-ichi Maezawa
  • Department of Mathematics, Tokyo University of Science, Japan
Yuta Nozaki
  • Faculty of Environment and Information Sciences, Yokohama National University, Japan
  • SKCM, Hiroshima University, Japan
Yoshio Okamoto
  • Graduate School of Informatics and Engineering, The University of Electro-Communications, Tokyo, Japan
Kenta Ozeki
  • Faculty of Environment and Information Sciences, Yokohama National University, Japan

Funding

  • Ito, Takehiro: JSPS KAKENHI Grant Numbers JP18H04091, JP19K11814, JP20H05793.
  • Iwamasa, Yuni: JSPS KAKENHI Grant Numbers JP20K23323, JP20H05795, JP22K17854.
  • Kakimura, Naonori: JSPS KAKENHI Grant Numbers JP20H05795, JP21H03397, JP22H05001.
  • Kobayashi, Yusuke: JSPS KAKENHI Grant Numbers JP20K11692, JP20H05795, JP22H05001.
  • Maezawa, Shun-ichi: JSPS KAKENHI Grant Numbers JP20H05795, JP22K13956.
  • Nozaki, Yuta: JSPS KAKENHI Grant Numbers JP20H05795, JP20K14317, JP23K12974.
  • Okamoto, Yoshio: JSPS KAKENHI Grant Numbers JP20H05795, JP20K11670, JP23K10982.
  • Ozeki, Kenta: JSPS KAKENHI Grant Numbers JP18K03391, JP19H01803, JP20H05795, 23K03195.

Acknowledgements

We thank Naoyuki Kamiyama for the discussion and the anonymous referees for their helpful comments.

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