  Max-product Belief Propagation (BP) is a popular message-passing algorithm
for computing a Maximum-A-Posteriori (MAP) assignment over a distribution
represented by a Graphical Model (GM). It has been shown that BP can solve a
number of combinatorial optimization problems including minimum weight
matching, shortest path, network flow and vertex cover under the following
common assumption: the respective Linear Programming (LP) relaxation is tight,
i.e., no integrality gap is present. However, when LP shows an integrality gap,
no model has been known which can be solved systematically via sequential
applications of BP. In this paper, we develop the first such algorithm, coined
Blossom-BP, for solving the minimum weight matching problem over arbitrary
graphs. Each step of the sequential algorithm requires applying BP over a
modified graph constructed by contractions and expansions of blossoms, i.e.,
odd sets of vertices. Our scheme guarantees termination in O(n^2) of BP runs,
where n is the number of vertices in the original graph. In essence, the
Blossom-BP offers a distributed version of the celebrated Edmonds' Blossom
algorithm by jumping at once over many sub-steps with a single BP. Moreover,
our result provides an interpretation of the Edmonds' algorithm as a sequence
of LPs.
