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Inference tasks in Markov random fields (MRFs) are closely related to the constraint satisfaction problem (CSP) and its soft generalizations. In particular, MAP inference in MRF is equivalent to the weighted (maxsum) CSP. A well-known tool to tackle CSPs are arc consistency algorithms, a.k.a. relaxation labeling. A promising approach to MAP inference in MRFs is linear programming relaxation solved by sequential treereweighted message passing (TRW-S). There is a not widely known algorithm equivalent to TRW-S, max-sum diffusion, which is slower but very simple. We give two theoretical results. First, we show that arc consistency algorithms and max-sum diffusion become the same thing if formulated in an abstractalgebraic way. Thus, we argue that max-sum arc consistency algorithm or max-sum relaxation labeling is a more suitable name for max-sum diffusion. Second, we give a criterion that strictly decreases during these algorithms. It turns out that every class of equivalent problems contains a unique problem that is minimal w.r.t. this criterion.
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