  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.
