Solving Constrained Combinatorial Optimisation Problems via MAP Inference without High-Order Penalties

Solving constrained combinatorial optimisation problems via MAP inference is often achieved by introducing extra potential functions for each constraint. This can result in very high order potentials, e.g. a 2nd-order objective with pairwise potentials and a quadratic constraint over all N variables would correspond to an unconstrained objective with an order-N potential. This limits the practicality of such an approach, since inference with high order potentials is tractable only for a few special classes of functions. We propose an approach which is able to solve constrained combinatorial problems using belief propagation without increasing the order. For example, in our scheme the 2nd-order problem above remains order 2 instead of order N. Experiments on applications ranging from foreground detection, image reconstruction, quadratic knapsack, and the M-best solutions problem demonstrate the effectiveness and efficiency of our method. Moreover, we show several situations in which our approach outperforms commercial solvers like CPLEX and others designed for specific constrained MAP inference problems.