Revisiting the Linear Programming Relaxation Approach to Gibbs Energy Minimization and Weighted Constraint Satisfaction

Abstract

Abstract-We present a number of contributions to the LP relaxation approach to weighted constraint satisfaction (= Gibbs energy minimization). We link this approach to many works from constraint programming, which relation has so far been ignored in machine vision and learning. While the approach has been mostly considered only for binary constraints, we generalize it to n-ary constraints in a simple and natural way. This includes a simple algorithm to minimize the LP-based upper bound, n-ary max-sum diffusionhowever, we consider using other bound-optimizing algorithms as well. The diffusion iteration is tractable for a certain class of higharity constraints represented as a black-box, which is analogical to propagators for global constraints CSP. Diffusion exactly solves permuted n-ary supermodular problems. A hierarchy of gradually tighter LP relaxations is obtained simply by adding various zero constraints and coupling them in various ways to existing constraints. Zero constraints can be added incrementally, which leads to a cutting plane algorithm. The separation problem is formulated as finding an unsatisfiable subproblem of a CSP.

Partial Enumeration and Curvature Regularization

“…Many methods have been proposed in the literature for solving higher-order relaxations, e.g. [29,15,19,33,14] to name just a few. To understand the relation to these methods, in [22] we analyze which specific relaxation is solved by our approach.…”

Nonlinearly Constrained MRFs: Exploring the Intrinsic Dimensions of Higher-Order Cliques

“…At this point, we have presented a complete algorithmic framework for solving the NC-MRF inference problem of Eq. 4 based on the common structure of exiting dualoptimization based approaches (e.g., [8,9,10,25]). Since it is unlikely (unless P = N P) that any algorithm can obtain a guaranteed approximate solution to the general MRF optimization problem of Eq.…”

“…Nevertheless, it remains challenging to even approximate the maximum a posteriori (MAP) configurations on graphs with large clique size, since most MAP inference algorithms scale exponentially with the size of the maximal clique in the graph [21,25] (in the general case, even encoding the higher-order potentials would require exponentially large space with respect to the clique size). To overcome such a limitation, previous methods usually exploit tractable structures in higherorder cliques by representing the higher-order potentials us- ing simpler functions [6,7,9,16,18].…”

The Complexity of Valued Constraint Satisfaction Problems

“…Some other approaches to using linear programming for VCSPs have been studied in [73,75] and in the context of computer vision [268,269]. In particular, as the dual of the so-called optimal soft arc consistency (OSAC) [75] is a tighter relaxation than BLP (see [264] for details), OSAC solves all problems solved by BLP.…”