The application of decision diagrams in combinatorial optimization has proliferated in the last decade. In recent years, authors have begun to investigate how to utilize not one, but a set of diagrams, to model constraints and objective function terms. Optimizing over a collection of decision diagrams, the problem we refer to as the consistent path problem (CPP), can be addressed by associating a network-flow model with each decision diagram, jointly linked through channeling constraints. A direct application of integer programming to the ensuing model has already shown to result in algorithms that provide orders-of-magnitude performance gains over classical methods. Lacking, however, is a careful study of dedicated solution methods designed to solve the CPP. This paper provides a detailed study of the CPP, including a discussion on complexity results and a complete polyhedral analysis. We propose a cut-generation algorithm which, under a structured ordering property, finds a cut, if one exists, through an application of the classical maximum flow problem, albeit in an exponentially sized network. We use this procedure to fuel a cutting-plane algorithm that is applied to unconstrained binary cubic optimization and a variant of the market split problem, resulting in a state-of-the-art algorithm for both.
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Institution: Clemson University, Department of Industrial Engineering, 100B Freeman Hall, Clemson, SC 29634