Multiple-choice Knapsack Constraint in Graphical Models

Abstract

Graphical models, such as cost function networks (CFNs), can compactly express large decomposable functions, which leads to efficient inference algorithms. Most methods for computing lower bounds in Branch-and-Bound minimization compute feasible dual solutions of a specific linear relaxation. These methods are more effective than solving the linear relaxation exactly, with better worst-case time complexity and better performance in practice. However, these algorithms are specialized to the structure of the linear relaxation of a CFN and cannot, for example, deal with constraints that cannot be expressed in extension, such as linear constraints of large arity. In this work, we show how to extend soft local consistencies, a set of approximate inference techniques for CFNs, so that they handle linear constraints, as well as combinations of linear constraints with at-mostone constraints. We embedded the resulting algorithm in toulbar2, an exact Branch-and-Bound solver for CFNs which has demonstrated superior results in several graphical model competitions and is state-of-theart for solving large computational protein design (CPD) problems. We significantly improved performance of the solver in CPD with diversity guarantees. It also compared favorably with integer linear programming solvers on knapsack problems with conflict graphs.

Bi-objective Discrete Graphical Model Optimization

“…When adding the higher-order GM bounding constraint, after having tried to decompose all non-unary cost functions into a sum of unary cost functions [17], we check if N contains only unary cost functions. In this case, the objective is linear and the higher-order GM bounding constraint can be replaced by two generalized linear constraints, without introducing any extra variables [30].…”