The β-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 β-assignment of G with respect to U is an edge set X such that degx(ν)=1 for all vertices ν in U, where degx(ν) is the degree of ν in the subgraph of G induced by the edge set X. The β-assignment problem is to find a β-assignment X such that β(X)≡max*degx(ν):νεV − U* is minimum. The purpose of this paper is to give an O(n3)-time algorithm for the β-assignment problem in general graphs. As byproducts, we also obtain a duality theorem as well as a necessary and sufficient condition for the existence of a β-assignment for a general graph. The latter result is a generalization of Tutteʹs theorem for the existence of a perfect matching of a general graph.