Stochastic Matching in Hypergraphs

We study the stochastic matching problem on k-uniform hypergraphs. In this problem, we are given a hypergraphH on V vertices. With each hyperedge E, we are given a probability pe of its existence i.e. an edge e exists in the hypergraph with probability pe , independent of all other edges. Each edge also has a weight associated with it. Our goal is to construct a matching with maximum expected weight. The only way we can find out if the an edge e exists or not is to try to query it, and if the edge is indeed present in the graph, we must include it in our matching. How should we adaptively query the edge set so that we attain our goal? We present an LP based 1 k+1/2 + o(1/k) approximation algorithm for this problem. This improves on the current best known bound of k + 1 due to Bansal et al. [1]. We use the standard LP relaxation for the hypergraph matching problem to get an upper bound on the optimal algorithm. The LP variables give us a hint on the probabilities with which to probe edges. We then present a new attenuation scheme which “attenuates” the edge probabilities given by the LP to get a good approximation ratio. Intuitively, attenuation is needed because if an edge is probed with very high probability, it hurts some of the other edges. Instead of attenuating all the edges by the same factor (as in Bansal et al. [1]), our scheme attenuates each edge differently.