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Reachability of Independent Sets and Vertex Covers Under Extended Reconfiguration Rules

Authors Shuichi Hirahara , Naoto Ohsaka , Tatsuhiro Suga , Akira Suzuki , Yuma Tamura , Xiao Zhou

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LIPIcs.ISAAC.2025.39.pdf
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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Problems, reductions and completeness
Keywords
  • combinatorial reconfiguration
  • extended reconfiguration rule
  • independent set reconfiguration
  • vertex cover reconfiguration
  • PSPACE-completeness
  • NP-completeness

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Abstract

In reconfiguration problems, we are given two feasible solutions to a graph problem and asked whether one can be transformed into the other via a sequence of feasible intermediate solutions under a given reconfiguration rule. While earlier work focused on modifying a single element at a time, recent studies have started examining how different rules impact computational complexity.
Motivated by recent progress, we study Independent Set Reconfiguration (ISR) and Vertex Cover Reconfiguration (VCR) under the k-Token Jumping (k-TJ) and k-Token Sliding (k-TS) models. In k-TJ, up to k vertices may be replaced, while k-TS additionally requires a perfect matching between removed and added vertices. It is known that the complexity of ISR crucially depends on k, ranging from PSPACE-complete and NP-complete to polynomial-time solvable.
In this paper, we further explore the gradient of computational complexity of the problems. We first show that ISR under k-TJ with k = |I| - μ remains NP-hard when μ is any fixed positive integer and the input graph is restricted to graphs of maximum degree 3 or planar graphs of maximum degree 4, where |I| is the size of feasible solutions. In addition, we prove that the problem belongs to NP not only for μ = O(1) but also for μ = O(log |I|). In contrast, we show that VCR under k-TJ is in XP when parameterized by μ = |S| - k, where |S| is the size of feasible solutions. Furthermore, we establish the PSPACE-completeness of ISR and VCR under both k-TJ and k-TS on several graph classes, for fixed k as well as superconstant k relative to the size of feasible solutions.

Cite As Get BibTex

Shuichi Hirahara, Naoto Ohsaka, Tatsuhiro Suga, Akira Suzuki, Yuma Tamura, and Xiao Zhou. Reachability of Independent Sets and Vertex Covers Under Extended Reconfiguration Rules. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 39:1-39:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ISAAC.2025.39

Author Details

Shuichi Hirahara
  • National Institute of Informatics, Tokyo, Japan
Naoto Ohsaka
  • CyberAgent, Inc., Tokyo, Japan
Tatsuhiro Suga
  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan
Akira Suzuki
  • Center for Data-Driven Science and Artificial Intelligence, Tohoku University, Sendai, Japan
Yuma Tamura
  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan
Xiao Zhou
  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan

Funding

This work was partially supported by Institute of Mathematics for Industry, Joint Usage/Research Center in Kyushu University. (FY2024 Workshop (II) "Theory of Combinatorial Reconfiguration and Beyond" (2024a037))
  • Suzuki, Akira: Partially supported by JSPS KAKENHI Grant Numbers JP25K14980.
  • Tamura, Yuma: Partially supported by JSPS KAKENHI Grant Number JP25K21148.

Acknowledgements

We are grateful to the anonymous referees for their helpful comments.

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