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# Markov Random Field MAP as Set Partitioning

; Proceedings of the Ninth International Conference on Probabilistic Graphical Models, PMLR 72:85-96, 2018.

#### Abstract

The Markov Random Field (MRF) MAP inference problem is considered from the viewpoint of integer programming (IP). The problem is shown to be a (pure) set partitioning problem (SPP). This allows us to bring existing work on SPP to bear on the MAP problem. Facets (maximally strong linear inequalities) of the closely related set packing (SP) problem are shown to be useful for MRF MAP. These facets include odd hole and odd anti-hole inequalities which are shown to be findable using a zero-half cut generator. Experimental results using CPLEX show that for MRF MAP problems, generating more zero-half cuts than normal typically brings performance improvements. Pre-processing methods to reduce the size of MRF MAP problems are also considered, and some preliminary results on their usefulness presented.