The ß-assignment problem in general graphs

We study a variation of the assignment problem in operations research and formulate it in terms of graphs as follows. Suppose G = (V; E) is a graph and U a subset of V. A-assignment of G with respect to U is an edge set X such that deg X (v) = 1 for all vertices v in U , where deg X (v) is the degree of v in the subgraph of G induced by the edge set X. The-assignment problem is to nd a-assignment X such that (X) maxfdeg X (v) : v 2 V ?Ug is minimum. The purpose of this paper is to give an O(n 3)-time algorithm for the-assignment problem in general graphs. As byproducts, we also get a duality theorem as well as a necessary and suucient condition for the existence of a-assignment for a general graph. The latter result is a generalization of Tutte's theorem for the existence of a perfect matching of a general graph.