Decision Diagram Relaxations for Integer Programming
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Mixed-integer programming (MIP) is often a practitioner’s primary approach when tackling hard discrete optimization problems. This important role was enabled by decades of theory and practical experience poured into modern MIP solvers. However, many problems are still challenging for MIP solvers, which motivates the need for novel perspectives to enhance MIP technology. In this dissertation, we explore the use of relaxed decision diagrams to improve MIP solvers. Relaxed decision diagrams are graph structures that encode relaxations of discrete optimization problems. One of their many uses in optimization is to generate bounds on the optimal value, which are strong in practice in the presence of certain types of structure. However, exporting decision diagram techniques to integer programming can be challenging due to the lack of clear structure to exploit. A first step is to develop methods to construct good relaxed decision diagrams from integer programming models. We propose a framework that focuses on a substructure of the problem and incorporates remaining constraints via Lagrangian relaxation and constraint propagation. In particular, we explore the use of a prevalent substructure in MIP solvers known as the conflict graph. Computational experiments indicate that they yield strong bounds under this framework. Once we understand how to build good relaxed decision diagrams, the next question is how to apply them in the context of a MIP solver. A MIP solver has several components, which include presolve, cutting planes, primal heuristics, and branch-and-bound. We show how relaxed decision diagrams can aid each of these components throughout the subsequent chapters. In particular, we view decision diagrams from a polyhedral perspective in order to generate cutting planes. Through a connection between decision diagrams and polarity, we develop an algorithm that generates cuts that are facet-defining for the convex hull of a decision diagram relaxation. We provide computational evidence that this algorithm generates strong cuts for structured problems. Finally, we investigate a broader integration of decision diagrams into MIP solvers. First, we show how relaxed decision diagrams can be used for bound and coefficient strengthening in the presolve step of a MIP solver. Second, we investigate the effect of applying cuts from decision diagrams throughout a branch-and-bound tree. Third, we generate both primal and dual bounds from decision diagrams to improve pruning within a branch-and-bound tree, which can result in significant improvements in solving time.