Counting Homomorphisms in Bipartite Graphs

Graph covers and the Bethe free energy have been useful theoretical tools for producing lower bounds on a variety of counting problems in graphical models, including the permanent and the ferromagnetic Ising model. Here, we propose a new conjecture that the Bethe free energy yields a lower bound on the weighted homomorphism counting problem over bipartite graphs. We show that this conjecture strengthens existing conjectures, and we prove the conjecture in several special cases using a novel reformulation of the graph cover characterization of the Bethe free energy.