Message passing for vertex covers

TL;DR

This paper introduces message passing algorithms, warning and survey propagation, as efficient heuristics for solving the NP-hard vertex cover problem, and explores their theoretical properties and extensions.

Contribution

It develops and analyzes message passing techniques for vertex cover, providing insights into their typical-case behavior and extending existing results.

Findings

Message passing algorithms effectively approximate minimal vertex covers.

Theoretical analysis recovers known results on random graphs.

Extensions of message passing improve understanding of solution space.

Abstract

Constructing a minimal vertex cover of a graph can be seen as a prototype for a combinatorial optimization problem under hard constraints. In this paper, we develop and analyze message passing techniques, namely warning and survey propagation, which serve as efficient heuristic algorithms for solving these computational hard problems. We show also, how previously obtained results on the typical-case behavior of vertex covers of random graphs can be recovered starting from the message passing equations, and how they can be extended.