Statistical properties and network structure of the marriage problem

We study the statistical properties of the distribution of total energies (or costs) over the set of N! configurations for the marriage problem with N men and N women. Analytical results and exhaustive numerical calculations for the whole configuration space are presented, taking into account several definitions of the individual energies. The distribution of the minimal energy is evaluated approximately. Moreover, we point out a natural association between the configuration space of the marriage problem and a network. We study the topological and metric properties of this network, as well as the distribution of local energy minima on it.