Dynamics of heuristic optimization algorithms on random graphs

TL;DR

This paper analyzes how heuristic algorithms for small vertex cover problems evolve on random graphs, modeling their dynamics as Markov processes and developing approximation methods validated by simulations.

Contribution

It introduces a Markovian framework for understanding heuristic graph algorithms and extends analysis to complex cases with new approximation schemes.

Findings

Markovian dynamics describe heuristic algorithms on random graphs

Approximation schemes match numerical simulations

Analysis includes solvable and complex cases

Abstract

In this paper, the dynamics of heuristic algorithms for constructing small vertex covers (or independent sets) of finite-connectivity random graphs is analysed. In every algorithmic step, a vertex is chosen with respect to its vertex degree. This vertex, and some environment of it, is covered and removed from the graph. This graph reduction process can be described as a Markovian dynamics in the space of random graphs of arbitrary degree distribution. We discuss some solvable cases, including algorithms already analysed using different techniques, and develop approximation schemes for more complicated cases. The approximations are corroborated by numerical simulations.