Asymptotically Optimal Inapproximability of Maxmin $k$-Cut Reconfiguration

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 prove that the optimal approximation factor of this problem is $1-\Theta \left(\frac{1}{k}\right)$ for every $k\ge 2$. Specifically, we show the $\mathsf{PSPACE}$-hardness of approximating the objective value within a factor of $1-\frac{\epsilon }{k}$ for some universal constant $\epsilon >0$, whereas we present a deterministic polynomial-time algorithm that achieves the approximation factor of $1-\frac{2}{k}$. To prove the hardness result, we develop a new probabilistic verifier that tests a ``striped'' pattern. Our polynomial-time algorithm is based on ``a random reconfiguration via a random solution,'' i.e., the transformation that goes through one random $k$-coloring.

arXiv comment63 pages, the first version including results for Maxmin k-Cut Reconfiguration and Maxmin Ek-SAT Reconfiguration has been divided into two papers