Asymptotically Optimal Inapproximability of Maxmin $k$-Cut Reconfiguration

TL;DR

This paper establishes the asymptotic inapproximability bounds for Maxmin k-Cut Reconfiguration, showing it is PSPACE-hard to approximate within certain factors, while providing a polynomial-time algorithm with a specific approximation guarantee.

Contribution

It proves the optimal approximation factor for Maxmin k-Cut Reconfiguration is 1 - Theta(1/k), establishing hardness and providing a matching approximation algorithm.

Findings

Hardness of approximation within 1 - ε/k for some ε > 0

Polynomial-time algorithm achieving 1 - 2/k approximation

Development of a probabilistic verifier using a striped pattern

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 $1 - \frac{\varepsilon}{k}$ 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.