Communities of Minima in Local Optima Networks of Combinatorial Spaces
This work addresses the problem of understanding configuration space structures for researchers in combinatorial optimization, though it is incremental as it builds on existing network analysis approaches.
The authors introduced a methodology to analyze the structure of configuration spaces in hard combinatorial problems by constructing networks of local optima and transitions between their basins, applying it to the quadratic assignment problem (QAP). They found that real-like instances exhibit clear modular structures in these networks, while random uniform instances are less partitionable, supported by statistical tests.
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.