Statistical Mechanics of the Hyper Vertex Cover Problem

TL;DR

This paper introduces the Hyper Vertex Cover problem, analyzes its phase diagram using statistical mechanics, and demonstrates efficient solution strategies via belief and survey propagation algorithms.

Contribution

It generalizes vertex cover to hypergraphs, provides explicit phase diagram analysis, and develops effective algorithms for large instances.

Findings

The HVC problem exhibits both replica symmetric and one-step replica symmetry breaking phases.

Explicit minimal density results are derived for different phase regimes.

Belief propagation and survey propagation algorithms efficiently solve large instances.

Abstract

We introduce and study a new optimization problem called Hyper Vertex Cover. This problem is a generalization of the standard vertex cover to hypergraphs: one seeks a configuration of particles with minimal density such that every hyperedge of the hypergraph contains at least one particle. It can also be used in important practical tasks, such as the Group Testing procedures where one wants to detect defective items in a large group by pool testing. Using a Statistical Mechanics approach based on the cavity method, we study the phase diagram of the HVC problem, in the case of random regualr hypergraphs. Depending on the values of the variables and tests degrees different situations can occur: The HVC problem can be either in a replica symmetric phase, or in a one-step replica symmetry breaking one. In these two cases, we give explicit results on the minimal density of particles, and the structure of the phase space. These problems are thus in some sense simpler than the original vertex cover problem, where the need for a full replica symmetry breaking has prevented the derivation of exact results so far. Finally, we show that decimation procedures based on the belief propagation and the survey propagation algorithms provide very efficient strategies to solve large individual instances of the hyper vertex cover problem.