Tightening MRF Relaxations with Planar Subproblems

We describe a new technique for computing lower-bounds on the minimum energy configuration of a planar Markov Random Field (MRF). Our method successively adds large numbers of constraints and enforces consistency over binary projections of the original problem state space. These constraints are represented in terms of subprob-lems in a dual-decomposition framework that is optimized using subgradient techniques. The complete set of constraints we consider enforces cycle consistency over the original graph. In practice we find that the method converges quickly on most problems with the addition of a few subproblems and outperforms existing methods for some interesting classes of hard potentials. 1 Introduction A standard approach to finding maximum a poste-riori (MAP) solutions (or equivalently minimum energy configurations) in a pairwise Markov random field (MRF) is to relax the combinatorial problem to a linear program while enforcing constraints that try to assure integrality of the resulting solution. The chief difficulty is that there are a huge number of possible constraints and only a small subset can possibly be enforced. The best understood case is that of imposing consistency constraints on each pair of variables along an edge.