Maxmin $k$-Cut 重构的渐近最优不可近似性
Asymptotically Optimal Inapproximability of Maxmin $k$-Cut Reconfiguration
AI总结：
本文证明 Maxmin $k$-Cut 重构问题的最优近似因子为 $1 - \Theta(1/k)$，并给出 PSPACE-困难性下界与多项式时间算法上界。
摘要：
$k$-着色重构（$k$-Coloring Reconfiguration）是研究最充分的重构问题之一，它要求通过反复重新着色单个顶点，将图的一个给定正常 $k$-着色转换为另一个正常 $k$-着色。其近似版本 Maxmin $k$-Cut 重构被定义为一个优化问题，即在（不一定是正常的）$k$-着色之间的转换过程中，最大化双色边的最小比例。在本文中，我们证明对于每个 $k \ge 2$，该问题的最优近似因子为 $1 - \Theta\left(\frac{1}{k}\right)$。具体而言，我们证明了对于某个通用常数 $\varepsilon > 0$，在 $1 - \frac{\varepsilon}{k}$ 因子内近似目标值是 $\mathsf{PSPACE}$-困难的，同时我们给出一个确定性多项式时间算法，其近似因子为 $1 - \frac{2}{k}$。为了证明困难性结果，我们开发了一个新的概率验证器来测试“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构”，即经过一个随机 $k$-着色的转换。
英文摘要：
$k$-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 $1 - Θ\left(\frac{1}{k}\right)$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
