High-arity interactions, polyhedral relaxations, and cutting plane algorithm for soft constraint optimisation (MAP-MRF)

LP relaxation approach to soft constraint optimisation (i.e. MAP-MRF) has been mostly considered only for binary problems. We present its generalisation to n-ary problems, including a simple algorithm to optimise the LP bound, n-ary max-sum diffusion. As applications, we show that a hierarchy of gradually tighter polyhedral relaxations of MAP-MRF is obtained by adding zero interactions. We propose a cutting plane algorithm, where cuts correspond to adding zero interactions and the separation problem to finding an unsatisfiable constraint satisfaction subproblem. Next, we show that certain high-arity interactions, e.g. certain global constraints, can be included into the framework in a principled way. Finally, we prove that n-ary max-sum diffusion finds global optimum for n-ary supermodular problems.