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In this work we present a new methodology to study the structure of theconfiguration spaces of hard combinatorial problems. It consists in buildingthe network that has as nodes the locally optimal configurations and as edgesthe weighted oriented transitions between their basins of attraction. We applythe approach to the detection of communities in the optima networks produced bytwo different classes of instances of a hard combinatorial optimizationproblem: the quadratic assignment problem (QAP). We provide evidence indicatingthat the two problem instance classes give rise to very different configurationspaces. For the so-called real-like class, the networks possess a clear modularstructure, while the optima networks belonging to the class of random uniforminstances are less well partitionable into clusters. This is convincinglysupported by using several statistical tests. Finally, we shortly discuss theconsequences of the findings for heuristically searching the correspondingproblem spaces.

Communities of Minima in Local Optima Networks of Combinatorial Spaces