Integration of AI and OR Techniques in Constraint Programming: 12th International Conference, CPAIOR 2015, Barcelona, Spain, May 18-22, 2015, Proceedings

s of Fast Tracked Journal Papers Lagrangian Bounds from Decision Diagrams David Bergman, Andre A. Cire, and Willem-Jan van Hoeve 1 School of Business, University of Connecticut, USA 2 Department of Management, University of Toronto Scarborough, Canada 3 Tepper School of Business, Carnegie Mellon University, USA david.bergman@business.uconn.edu,acire@utsc.utoronto.ca, vanhoeve@andrew.cmu.edu Decision diagrams are compact graphical representations of Boolean functions widely applied in circuit design and verification. Recently, relaxed decision diagrams were introduced as a new type of relaxation for discrete optimization problems [2]. In the context of constraint programming, relaxed decision diagrams have been successfully applied to improve constraint propagation and optimization reasoning [1,3]. One typically associates a decision diagram with a specific global constraint that is defined on a subset of variables (its scope), and the diagram is subsequently filtered and refined according to the other constraints of the problem. Additionally, if the objective function is evaluated on the same set of variables, the decision diagram can be used to obtain optimization bounds and for applying cost-based filtering. In this work we propose a technique to strengthen a relaxed decision diagram of a problem by incorporating inference from constraints via a Lagrangian relaxation method. Namely, we associate penalties with the constraints that may be potentially violated by the solutions encoded in a relaxed decision diagram. These penalties are incorporated directly into the diagram as arc costs, which are taken into account in the diagram’s objective function evaluation. We show that, with this generic approach, the resulting diagram may potentially yield stronger optimization bounds than the one obtained from the original relaxation, while the associated cost-based filtering allows for further refining the diagram and ultimately reducing the search space. If the incorporated constraints are linear, we also demonstrate that the optimal set of penalties are the duals of a shortest-path linear program derived from the decision diagram. To evaluate our approach, we perform computational experiments on the traveling salesman problem with time windows. Relaxed decision diagrams are used in a global constraint enforcing that city visits must not overlap in time, and tests are performed with a state-of-the-art constraint-based scheduler. Results show that the diagram with improved Lagrangian bounds can drastically reduce solution times in comparison to the original relaxation.