Communities of Minima in Local Optima Networks of Combinatorial Spaces

TL;DR

This paper introduces a new network-based methodology to analyze the structure of local optima in combinatorial problems, revealing distinct community structures in different instance classes of the quadratic assignment problem.

Contribution

It presents a novel approach to study configuration spaces via optima networks and demonstrates its effectiveness on different classes of QAP instances, highlighting structural differences.

Findings

Real-like instances have clear community structures in their optima networks.

Random uniform instances show less modularity in their optima networks.

The structural differences have implications for heuristic search strategies.

Abstract

In this work we present a new methodology to study the structure of the configuration spaces of hard combinatorial problems. It consists in building the network that has as nodes the locally optimal configurations and as edges the weighted oriented transitions between their basins of attraction. We apply the approach to the detection of communities in the optima networks produced by two different classes of instances of a hard combinatorial optimization problem: the quadratic assignment problem (QAP). We provide evidence indicating that the two problem instance classes give rise to very different configuration spaces. For the so-called real-like class, the networks possess a clear modular structure, while the optima networks belonging to the class of random uniform instances are less well partitionable into clusters. This is convincingly supported by using several statistical tests. Finally, we shortly discuss the consequences of the findings for heuristically searching the corresponding problem spaces.