Max Product for Max-Weight Independent Set and Matching

The Max Product (MP) is a local, iterative, message passing style algorithm that has been developed for finding the maximum a posteriori (MAP) assignment of discrete probability distribution specified by a graphical model. The scope of application of MP is vast and in particular it can serve as a heuristic to solve any combinatorial optimization problem. Despite the success of MP algorithm in the context of coding and vision, not much has been theoretically understood about the correctness and convergence of MP. The Maximum Weight Independent Set (MWIS) and Maximum Weight Matching (MWM) are classically well studied combinatorial optimization problems. A lot of work has been done to design efficient algorithms for finding MWIS and MWM. In this paper, we study application of MP algorithm for MWIS and MWM for sparse random graphs: G(n, c/n) and Gr(n), which are n node random graphs with parameter c and r respectively. We show that when weights (node or edge depending on MWIS or MWM) are assigned independently according to exponential distribution, the MP algorithm converges and finds correct solution for a large range of parametric value c and r. In particular, we show that for any ǫ > 0, for large enough n, the MP becomes 1 + ǫ competitive with probability at least 1 − ǫ. Our results build upon the results of Gamarnik, Nowicki and Swirscsz (2005), which established local optimality property of MWIS and MWM for sparse random graphs.