The Demand Matching Problem

We examine formulations for the well-known b-matching problem in the presence of integer demands on the edges. A subset M of edges is feasible if for each node v, the total demand of edges in M incident to it is at most b_v. We examine the system of star inequalities for this problem. This system yields an exact linear description for b-matchings in bipartite graphs. For the demand version, we show that the integrality gap for this system is at least 2 1/2 and at most 2 13/16. For general graphs, the gap lies between 3 and 3 5/16. A fully polynomial approximation scheme is also presented for the problem on a tree, thus generalizing a well-known result for the knapsack problem.