Expected Approximation Guarantees for the Demand Matching Problem

The objective of the demand matching problem is to obtain the subset M of edges which is feasible and where the sum of the profits of each of the edges is maximized. The set M is feasible if for each vertex v the total demand of edges in M incident to v is at most bv. In the case where each of the edges has one unit profit, the problem becomes finding the subset of largest size and hence, is called the cardinality problem. Shepherd and Vetta [SV06] demonstrate that the integrality gap for the general demand matching problem for bipartite graphs is between 2:5 and 2:764, and between 3 and 3:264 non-bipartite graphs. We demonstrate that an expected 2:5-approximation guarantee and 3-approximation guarantee is achieveable for bipartite graphs and non-bipartite graphs and give some connections to the independent set and weighted independent set problem.