On the exactness of the cavity method for Weighted b-Matchings on   Arbitrary Graphs and its Relation to Linear Programs
  We consider the general problem of finding the minimum weight b-matching on arbitrary graphs. We prove that, whenever the linear programming relaxation of the problem has no fractional solutions, then the cavity or belief propagation equations converge to the correct solution both for synchronous and asynchronous updating. 
M. Bayati, C. Borgs, J. Chayes, R. Zecchina