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A Linear Programming Approach to Max-Sum Problem: A Review
July 2007 (vol. 29 no. 7)
pp. 1165-1179
Tom? Werner, IEEE Computer Society
The max-sum labeling problem, defined as maximizing a sum of binary (i.e., pairwise) functions of discrete variables, is a general NP-hard optimization problem with many applications, such as computing the MAP configuration of a Markov random field. We review a not widely known approach to the problem, developed by Ukrainian researchers Schlesinger et al. in 1976, and show how it contributes to recent results, most importantly, those on the convex combination of trees and tree-reweighted max-product. In particular, we review Schlesinger et al.'s upper bound on the max-sum criterion, its minimization by equivalent transformations, its relation to the constraint satisfaction problem, the fact that this minimization is dual to a linear programming relaxation of the original problem, and the three kinds of consistency necessary for optimality of the upper bound. We revisit problems with Boolean variables and supermodular problems. We describe two algorithms for decreasing the upper bound. We present an example application for structural image analysis.
Index Terms:
Markov random fields, undirected graphical models, constraint satisfaction, belief propagation, linear programming relaxation, max-sum, max-plus, max-product, supermodular optimization.
Citation:
Tom? Werner, "A Linear Programming Approach to Max-Sum Problem: A Review," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 29, no. 7, pp. 1165-1179, July 2007, doi:10.1109/TPAMI.2007.1036
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