number CTU – CMP – 2007 – 27 September 13 , 2008

We address the problem of energy minimization, which is (1) generally NP-complete and (2) involves many discrete variables – commonly a 2D array of them, arising from an MRF model. One of the approaches to the problem is to formulate it as integer linear programming and relax integrality constraints. However this can be done in a number of possible ways. One, widely applied previously (LP-1) [19, 13, 4, 22, 9, 23], appears to lead to a large-scale linear program which is not practical to solve with general LP methods. A number of algorithms were developed which attempt to solve the problem exploiting its structure [14, 23, 22, 9], however their common drawback is that they may converge to a suboptimal point. The other LP relaxation we consider here is constructed by (1) reformulating the optimization problem in the form of a function of binary variables [18], and (2) applying the roof duality relaxation [6] to the reformulated problem. We refer to the resulting relaxation as LP-2. It is different from LP-1 in many respects, and this is the main question of our study. Most importantly, it is possible to apply an efficient, fully combinatorial, algorithm to solve the relaxed problem. We also derive the following relations: a) LP-1 is generally a tighter relaxation than LP-2, b) LP-2 provides constraints on optimal integer configurations, which allows one to identify “a part” of an optimal solution, c) a subclass of problems can be identified for which LP-2 is as tight as LP-1 providing additional characterization of solutions of LP-1 for this subclass. Our last contribution is providing an alternative formulation of LP-2: we prove that it is equivalent to computing a decomposition of the energy into submodular and supermodular parts so that the sum of the lower bounds for each part is maximized.