Relaxation-Aware Heuristics for Exact Optimization in Graphical Models.

Exact solvers for optimization problems on graphical models, such as Cost Function Networks and Markov Random Fields, typically use branch-and-bound. The efficiency of the search relies mainly on two factors: the quality of the bound computed at each node of the branchand-bound tree and the branching heuristics. In this respect, there is a trade-off between quality of the bound and computational cost. In particular, the Virtual Arc Consistency (VAC) algorithm computes high quality bounds but at a significant cost, so it is mostly used in preprocessing, rather than in every node of the search tree. In this work, we identify a weakness in the use of VAC in branch-and-bound solvers, namely that they ignore the information that VAC produces on the linear relaxation of the problem, except for the dual bound. In particular, the branching heuristic may make decisions that are clearly ineffective in light of this information. By eliminating these ineffective decisions, we significantly reduce the size of the branch-and-bound tree. Moreover, we can optimistically assume that the relaxation is mostly correct in the assignments it makes, which helps find high quality solutions quickly. The combination of these methods shows great performance in some families of instances, outperforming the previous state of the art.