Minimum vertex cover problems on random hypergraphs: replica symmetric solution and a leaf removal algorithm

TL;DR

This paper investigates the minimum vertex cover problem on random hypergraphs using statistical mechanics and an algorithm, revealing a phase transition at a critical average degree that affects solution accuracy and method reliability.

Contribution

It introduces a combined analysis using replica symmetry and leaf removal algorithms to understand phase transitions in hypergraph vertex cover problems.

Findings

Phase transition at average degree e/(1)-1

Replica symmetric solution valid below critical degree

Algorithm matches replica method below critical degree

Abstract

We study minimum vertex cover problems on random \alpha-uniform hypergraphs using two different approaches, a replica method in statistical mechanics of random systems and a leaf removal algorithm. It is found that there exists a phase transition at the critical average degree e/(\alpha-1). Below the critical degree, a replica symmetric ansatz in the statistical-mechanical method holdsand the algorithm estimates a solution of the problem which coincide with that by the replica method. In contrast, above the critical degree, the replica symmetric solution becomes unstable and these methods fail to estimate the exact solution.These results strongly suggest a close relation between the replica symmetry and the performance of approximation algorithm.