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Independent Set Reconfiguration on Directed Graphs

Authors Takehiro Ito , Yuni Iwamasa , Yasuaki Kobayashi , Yu Nakahata , Yota Otachi , Masahiro Takahashi, Kunihiro Wasa

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Author Details

Takehiro Ito
  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan
Yuni Iwamasa
  • Graduate School of Informatics, Kyoto University, Kyoto, Japan
Yasuaki Kobayashi
  • Graduate School of Information Science and Technology, Hokkaido University, Sapporo, Japan
Yu Nakahata
  • Division of Information Science, Nara Institute of Science and Technology, Ikoma, Japan
Yota Otachi
  • Graduate School of Informatics, Nagoya University, Nagoya, Japan
Masahiro Takahashi
  • Graduate School of Informatics, Kyoto University, Kyoto, Japan
Kunihiro Wasa
  • Faculty of Science and Engineering, Hosei University, Tokyo, Japan

Funding

  • Ito, Takehiro: JSPS KAKENHI Grant Numbers JP18H04091, JP19K11814, JP20H05793
  • Iwamasa, Yuni: JSPS KAKENHI Grant Numbers JP20K23323, JP20H05795, JP22K17854
  • Kobayashi, Yasuaki: JSPS KAKENHI Grant Numbers JP20H00595, JP20H05793, JP20K19742
  • Nakahata, Yu: JSPS KAKENHI Grant Numbers JP19H01103, JP20H05794, JP22H03549, JP22K17851
  • Otachi, Yota: JSPS KAKENHI Grant Numbers JP18H04091, JP20H05793, JP21K11752, JP22H00513
  • Wasa, Kunihiro: JSPS KAKENHI Grant Numbers JP19K20350, JP20H05793, JP22H03549, JP22K17849

Cite As Get BibTex

Takehiro Ito, Yuni Iwamasa, Yasuaki Kobayashi, Yu Nakahata, Yota Otachi, Masahiro Takahashi, and Kunihiro Wasa. Independent Set Reconfiguration on Directed Graphs. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 58:1-58:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.MFCS.2022.58

Abstract

Directed Token Sliding asks, given a directed graph and two sets of pairwise nonadjacent vertices, whether one can reach from one set to the other by repeatedly applying a local operation that exchanges a vertex in the current set with one of its out-neighbors, while keeping the nonadjacency. It can be seen as a reconfiguration process where a token is placed on each vertex in the current set, and the local operation slides a token along an arc respecting its direction. Previously, such a problem was extensively studied on undirected graphs, where the edges have no directions and thus the local operation is symmetric. Directed Token Sliding is a generalization of its undirected variant since an undirected edge can be simulated by two arcs of opposite directions.
In this paper, we initiate the algorithmic study of Directed Token Sliding. We first observe that the problem is PSPACE-complete even if we forbid parallel arcs in opposite directions and that the problem on directed acyclic graphs is NP-complete and W[1]-hard parameterized by the size of the sets in consideration. We then show our main result: a linear-time algorithm for the problem on directed graphs whose underlying undirected graphs are trees, which are called polytrees. Such a result is also known for the undirected variant of the problem on trees [Demaine et al. TCS 2015], but the techniques used here are quite different because of the asymmetric nature of the directed problem. We present a characterization of yes-instances based on the existence of a certain set of directed paths, and then derive simple equivalent conditions from it by some observations, which yield an efficient algorithm. For the polytree case, we also present a quadratic-time algorithm that outputs, if the input is a yes-instance, one of the shortest reconfiguration sequences.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Combinatorial reconfiguration
  • token sliding
  • directed graph
  • independent set
  • graph algorithm

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