A generalization of Hungarian method and Hall's theorem with applications in wireless sensor networks

In this paper, we consider various problems concerning quasi-matchings and semimatchings in bipartite graphs, which generalize the classical problem of determining a perfect matching in bipartite graphs. We prove a vast generalization of Hall’s marriage theorem, and present an algorithm that solves the problem of determining a lexicographically minimum g-quasi-matching (that is a set F of edges in a bipartite graph such that in one set of the bipartition every vertex v has at least g(v) incident edges from F , where g is a so-called need mapping, while on the other side of the bipartition the distribution of degrees with respect to F is lexicographically minimum). We also present an application in designing an optimal CDMA-based wireless sensor networks.