Towards Improving Merging Heuristics for Binary Decision Diagrams

Over the last years, binary decision diagrams (BDDs) have become a powerful tool in the field of combinatorial optimization. They are directed acyclic multigraphs and represent the solution space of binary optimization problems in a recursive way. During their construction, merging of nodes in this multigraph is applied to keep the size within polynomial bounds resulting in a discrete relaxation of the original problem. The longest path length through this diagram corresponds then to an upper bound of the optimal objective value. The algorithm deciding which nodes to merge is called a merging heuristic. A commonly used heuristic for layer-wise construction is minimum longest path length (minLP) which sorts the nodes in a layer descending by the currently longest path length to them and subsequently merges the worst ranked nodes to reduce the width of a layer. A shortcoming of this approach is that it neglects the (dis-)similarity between states it merges, which we assume to have negative impact on the quality of the finally obtained bound. By means of a simple tie breaking procedure, we show a way to incorporate the similarity of states into minLP using different distance functions to improve dual bounds for the maximum independent set problem (MISP) and the set cover problem (SCP), providing empirical evidence for our assumption. Furthermore, we extend this procedure by applying similarity-based node merging also to nodes with close but not necessarily identical longest path values. This turns out to be beneficial for weighted problems where ties are substantially less likely to occur. We evaluate the method on the weighted MISP and tune parameters that control as to when to apply similarity-based node merging.