Statistical mechanics methods and phase transitions in optimization problems

TL;DR

This paper reviews how statistical mechanics methods are applied to analyze phase transitions and solution properties in combinatorial optimization problems, bridging physics and computer science.

Contribution

It introduces physicists' tools for optimization problems to computer scientists, focusing on phase transitions and statistical analysis in key combinatorial problems.

Findings

Analysis of phase transitions in combinatorial problems

Application of statistical mechanics to optimization

Insights into solution space structure

Abstract

Recently, it has been recognized that phase transitions play an important role in the probabilistic analysis of combinatorial optimization problems. However, there are in fact many other relations that lead to close ties between computer science and statistical physics. This review aims at presenting the tools and concepts designed by physicists to deal with optimization or decision problems in an accessible language for computer scientists and mathematicians, with no prerequisites in physics. We first introduce some elementary methods of statistical mechanics and then progressively cover the tools appropriate for disordered systems. In each case, we apply these methods to study the phase transitions or the statistical properties of the optimal solutions in various combinatorial problems. We cover in detail the Random Graph, the Satisfiability, and the Traveling Salesman problems. References to the physics literature on optimization are provided. We also give our perspective regarding the interdisciplinary contribution of physics to computer science.