A Linear Programming Approach to Max-sum Problem: A Review

Abstract

The max-sum labeling problem, de ned as maximizing a sum of functions of pairs of discrete variables, is a general optimization problem with numerous applications, e.g., computing MAP assignments of a Markov random eld. We review a not widely known approach to the problem based on linear programming relaxation, developed by Schlesinger et al. in 1976. We also show how this old approach contributes to more recent results, most importantly by Wainwright et al. In particular, we review Schlesinger's upper bound on the max-sum criterion, its minimization by equivalent transformations, its relation to constraint satisfaction problem, how it can be understood as a linear programming relaxation, and three kinds of consistency necessary for optimality of the upper bound. As special cases, we revisit problems with two labels and supermodular problems. We describe two algorithms for decreasing the upper bound. We present an example application to structural image analysis.