  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 gives inside into the complexity of the energy
landscape of different ensembles. Erd\"os-Renyi graphs and ring graphs with
randomly added edges are considered and two types of distributions for the
random edge weighs are used. For maximum-weight matching, fast and scalable
algorithms exist, such that we can study large graphs of more than $10^5$
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 here presented ensembles our results
allow for the presence of complex energy landscapes in the spirit of
Replica-Symmetry Breaking.
