We describe a new variational lower-bound on the minimum energy configuration
of a planar binary Markov Random Field (MRF). Our method is based on adding
auxiliary nodes to every face of a planar embedding of the graph in order to
capture the effect of unary potentials. A ground state of the resulting
approximation can be computed efficiently by reduction to minimum-weight
perfect matching. We show that optimization of variational parameters achieves
the same lower-bound as dual-decomposition into the set of all cycles of the
original graph.