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 semi-matchings 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.