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# Polynomial-Time Exact Inference in NP-Hard Binary MRFs via Reweighted Perfect Matching

*Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics*, PMLR 9:717-724, 2010.

#### Abstract

We develop a new form of reweighting (Wainwright et al., 2005b) to leverage the relationship between Ising spin glasses and perfect matchings into a novel technique for the exact computation of MAP states in hitherto intractable binary Markov random fields. Our method solves an $n \times n$ lattice with external field and random couplings much faster, and for larger $n$, than the best competing algorithms. It empirically scales as $O(n^3)$ even though this problem is NP-hard and non-approximable in polynomial time. We discuss limitations of our current implementation and propose ways to overcome them.