We review belief propagation algorithms inspired by the study of phase transitions in combinatorial optimization problems. In particular, we present rigorous results on convergence of such algorithms for matching and associated bargaining problems on networks. We also present a belief propagation algorithm for the prize- collecting Steiner tree problem, for which rigorous convergence results are not yet known. Finally, we show how this algorithm can be used to discover pathways in cancer genomics, and to suggest possible drug targets for cancer therapy. These methods give us the ability to share information across multiple patients to help reconstruct highly patient-specific networks.