Optimal Balanced Semi-Matchings for Weighted Bipartite Graphs

The matching of a bipartite graph is a structure that can be seen in various assignment problems and has long been studied. The semi-matching is an extension of the matching for a bipartite graph G =(U ∪ V, E). It is defined as a set of edges, M ⊆ E, such that each vertex in U is an endpoint of exactly one edge in M. The load-balancing problem is the problem of finding a semi-matching such that the degrees of each vertex in V are balanced. This problem is studied in the context of the task scheduling to find a “balanced” assignment of tasks for machines, and an O(¦E¦¦U¦) time algorithm is proposed. On the other hand, in some practical problems, only balanced assignments are not sufficient, e.g., the assignment of wireless stations (users)to access points (APs) in wireless networks. In wireless networks, the quality of the transmission depends on the distance between a user and its AP; shorter distances are more desirable. In this paper, We formulate the min-weight load-balancing problem of finding a balanced semi-matching that minimizes the total weight for weighted bipartite graphs. We then give an optimal condition of weighted semi-matchings and propose an O(¦E¦¦U¦¦V¦) time algorithm.