Energy landscapes of some matching-problem ensembles

The maximum-weight matching problem and the behavior of its energy landscape is numerically investigated. We apply a perturbation method adapted from the analysis of spin glasses. This method provides insight into the complexity of the energy landscape of different ensembles. Erdős–Rényi graphs and ring graphs with randomly added edges are considered, and two types of distributions for the random edge weights are used. Fast and scalable algorithms exist for maximum weight matching, allowing us to study large graphs with more than 10⁵ nodes. Our results show that the structure of the energy landscape for standard ensembles of matching is simple, comparable to the energy landscape of a ferromagnet. Nonetheless, for some of the ensembles presented here, our results allow for the presence of complex energy landscapes in the spirit of a replica-symmetry breaking scenario.