Universality of the Local Marginal Polytope

We show that solving the LP relaxation of the min-sum labeling problem (also known as MAP inference problem in graphical models, discrete energy minimization, or valued constraint satisfaction) is not easier than solving any linear program. Precisely, every polytope is linear-time representable by a local marginal polytope and every LP can be reduced in linear time to a linear optimization (allowing infinite costs) over a local marginal polytope. The reduction can be done (though with a higher time complexity) even if the local marginal polytope is restricted to have a planar structure.