Modeling and Exploiting Dominance Rules for Discrete Optimization with Decision Diagrams

Vianney Coppé, Xavier Gillard, Pierre Schaus

Abstract: Discrete optimization with decision diagrams is a recent approach to solve combinatorial problems that can be formulated with dynamic programming. It consists in a branch-and-bound algorithm that iteratively explores the search space by compiling bounded-width decision diagrams. Those decision diagrams are used both to subdivide a given problem into smaller subproblems – in a divide-and-conquer fashion – and to compute primal and dual bounds for those. It has been previously shown that pruning performed during the compilation of those decision diagrams can greatly impact the quality of the bounds, and consequently the performance of the branch-and-bound algorithm. In this paper, we study the integration of dominance rules inside the decision diagram-based optimization framework. We propose a modeling language for consistently formulating dominance rules for dynamic programming models, and describe how they can be exploited to systematically detect and prune dominated nodes during the search. Furthermore, we explain how to combine this additional filtering mechanism with caching techniques to further improve the performance of the algorithm. Dominance rules are shown to significantly reduce the number of nodes expanded and the running time of the algorithm on four optimization problems.