On the Consistent Path Problem

• 成果类型: Article
• 署名作者: Lozano, Leonardo; Bergman, David; Smith, J. Cole
• 署名单位: University System of Ohio; University of Cincinnati; University of Connecticut; Syracuse University
• 刊物名称: OPERATIONS RESEARCH
• ISSN/ISSBN: 0030-364X
• DOI: 10.1287/opre.2020.1979
• 发表日期: 2020
• 页码: 1913-1931
• 关键词: networks/graphs: flow algorithms; networks/graphs: generalized networks; programming: integer; algorithms

摘要

The application of decision diagrams in combinatorial optimization has proliferated in the last decade. In recent years, authors have begun to investigate how to use 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 been 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 an algorithm that compares favorably with CPLEX, using standard integer programming formulations for both problems.