Phase transition in the bipartite z-matching

We study numerically the maximum $z$-matching problems on ensembles of bipartite random graphs. The describes matching between two types nodes, users and servers, where each server may serve up to $z$ at same time. By using a mapping standard maximum-cardinality matching, because for latter there exists polynomial-time exact algorithm, we can large system sizes $10^6$ nodes. measure capacity energy resulting optimum matchings. First, confirm previous analytical results regular Next, finite-size behaviour find scaling as before which indicates universality problem. Finally, investigate Erd\H{o}s-R\'enyi graphs saturability function average degree, i.e., whether network allows many customers possible be served, i.e. exploiting servers in an optimal way. phase transitions unsaturable saturable phases. These coincide with strong change running time well point minimum-degree heuristic algorithm starts fail.