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Asymptotically Optimal Inapproximability of Maxmin k-Cut Reconfiguration

Authors Shuichi Hirahara , Naoto Ohsaka

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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Graph coloring
  • Mathematics of computing → Approximation algorithms
Keywords
  • reconfiguration problems
  • graph coloring
  • hardness of approximation

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Abstract

k-Coloring Reconfiguration is one of the most well-studied reconfiguration problems, which asks to transform a given proper k-coloring of a graph to another by repeatedly recoloring a single vertex. Its approximate version, Maxmin k-Cut Reconfiguration, is defined as an optimization problem of maximizing the minimum fraction of bichromatic edges during the transformation between (not necessarily proper) k-colorings. In this paper, we demonstrate that the optimal approximation factor of this problem is 1 - Θ(1/k) for every k ≥ 2. Specifically, we prove the PSPACE-hardness of approximating the objective value within a factor of 1 - ε/k for some universal constant ε > 0, whereas we develop a deterministic polynomial-time algorithm that achieves the approximation factor of 1 - 2/k.
To prove the hardness result, we propose a new probabilistic verifier that tests a "striped" pattern. Our approximation algorithm is based on a random transformation that passes through a random k-coloring.

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Shuichi Hirahara and Naoto Ohsaka. Asymptotically Optimal Inapproximability of Maxmin k-Cut Reconfiguration. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 96:1-96:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.96

Author Details

Shuichi Hirahara
  • National Institute of Informatics, Tokyo, Japan
Naoto Ohsaka
  • CyberAgent, Inc., Tokyo, Japan

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