$k$-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 $1 - \frac{\varepsilon}{k}$ for some universal constant $\varepsilon > 0$, 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.

翻译：$k$着色重配置是研究最深入的重配置问题之一，其要求通过反复对单个顶点重新着色，将图的一个给定真$k$着色变换为另一个真$k$着色。其近似版本——最大最小$k$割重配置——被定义为一个优化问题，目标是在（非必为真）$k$着色之间的变换过程中最大化双色边的最小比例。本文证明，对于所有$k \ge 2$，该问题的最优近似因子为$1 - \Theta\left(\frac{1}{k}\right)$。具体而言，我们证明了在$1 - \frac{\varepsilon}{k}$的近似因子内逼近目标值是$\mathsf{PSPACE}$难的，其中$\varepsilon > 0$为某通用常数；同时我们提出了一个确定性多项式时间算法，其能达到$1 - \frac{2}{k}$的近似因子。为证明该难度结果，我们开发了一种新的概率验证器，用于检测“条纹”模式。我们的多项式时间算法基于“通过随机解的随机重配置”，即经由一个随机$k$着色进行的变换过程。