Asymptotically Optimal Inapproximability of Maxmin $k$-Cut Reconfiguration

2024-10-04

AI Summary

This paper studies the Maxmin $k$-Cut Reconfiguration problem: given a graph, maximize the minimum fraction of bichromatic edges over all intermediate colorings along a reconfiguration path of proper $k$-colorings. We establish its asymptotically optimal approximation ratio as $1 - Theta(1/k)$. First, via a novel probabilistic verifier and intricate complexity analysis, we prove that for any constant $varepsilon > 0$, achieving a $(1 - varepsilon/k)$-approximation is PSPACE-hard. Second, we design the first deterministic polynomial-time algorithm attaining a $(1 - 2/k)$-approximation. Our results fully characterize the approximability threshold of the problem, simultaneously resolving its computational tractability boundary and providing an efficient constructive algorithm.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 $1 - Thetaleft(frac{1}{k} ight)$ 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.

Problem

• Optimal approximation for Maxmin k-Cut Reconfiguration
• PSPACE-hardness of approximating k-Coloring Reconfiguration
• Polynomial-time algorithm for k-Cut Reconfiguration

Innovation

• Developed a new probabilistic verifier
• Used random reconfiguration via random solution
• Achieved optimal approximation factor