Theoretical insights and algorithmic tools for decision diagram-based optimization
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The use of decision diagrams has recently emerged as a viable general solution approach for solving discrete optimization problems. The decision diagram data structure is used to explicitly represent, either exactly or approximately, the set of feasible solutions to a given problem. Techniques based on decision diagrams have been successfully used on a diverse set of applications, ranging from scheduling to combinatorial optimization, and have often outperformed commercial state-of-the-art constraint programming and integer programming technology. Lacking, however, is a thorough theoretical investigation into the quality of approximate decision diagrams, as well as the development of structured techniques for tightening relaxation bounds provided by approximate decision diagrams, analogously to how cutting-planes are used in integer programming. This paper provides an analysis of the strength of approximate decision diagrams, as well as the description of several bound-tightening procedures for problems with linear objective functions.
KeywordsDecision diagrams Discrete optimization Constraint satisfaction problems Constraint optimization problems
- 1.Andersen, H.R., Hadzic, T., Hooker, J.N., & Tiedemann, P. (2007). A constraint store based on multivalued decision diagrams. In Bessière, C. (Ed.) Principles and practice of constraint programming (CP 2007). Lecture notes in computer science, (Vol. 4741 pp. 118–132): Springer.Google Scholar
- 2.Becker, B., Behle, M., Eisenbrand, F., & Wimmer, R. (2005). BDDS in a branch and cut framework. In Nikoletseas, S. (Ed.) Experimental and efficient algorithms, proceedings of the 4th international workshop on efficient and experimental algorithms (WEA 05). Lecture notes in computer science, (Vol. 3503 pp. 452–463): Springer.Google Scholar
- 3.Behle, M. (2007). On threshold BDDs and the optimal variable ordering problem. In COCOA’07: Proceedings Of the 1st international conference on combinatorial optimization and applications (pp. 124–135). Berlin, Heidelberg: Springer.Google Scholar
- 6.Bergman, D., Cire, A.A., van Hoeve, W.J., & Hooker, J.N. (2015). Discrete optimization with decision diagrams. INFORMS Journal on Computing. to appear.Google Scholar
- 8.Bergman, D., Ciré, A.A., Sabharwal, A., Samulowitz, H., Saraswat, V.A., & van Hoeve, W.J. (2014). Parallel combinatorial optimization with decision diagrams. In Simonis, H. (Ed.) Integration of AI and OR Techniques in Constraint Programming - 11th International Conference, CPAIOR 2014, Cork, Ireland, May 19-23, 2014. Proceedings. Lecture Notes in Computer Science. doi:10.1007/978-3-319-07046-9_25, (Vol. 8451 pp. 351–367) Springer.
- 10.Bergman, D., van Hoeve, W.J., & Hooker, J.N. (2011). Manipulating MDD relaxations for combinatorial optimization. In Achterberg, T., & Beck, J.C. (Eds.) CPAIOR. Lecture notes in computer science, (Vol. 6697 pp. 20–35): Springer.Google Scholar
- 11.Bergman, D. (2013). New techniques for discrete optimization. Ph.D. thesis, Carnegie Mellon University.Google Scholar
- 14.Cire, A.A., & van Hoeve, W.J. (2012). MDD Propagation for disjunctive scheduling. In Proceedings of the twenty-second international conference on automated planing and scheduling (ICAPS) (pp. 1–1): AAAI Press.Google Scholar
- 17.Gopalan, P., Klivans, A., Meka, R., Stefankovic, D., Vempala, S., & Vigoda, E. (2011). An fptas for #knapsack and related counting problems. In IEEE 52nd annual symposium on Foundations of computer science (FOCS), 2011 (pp. 817–826).Google Scholar
- 18.Hadzic, T., & Hooker, J.N. (2007). Cost-bounded binary decision diagrams for 0-1 programming. In Loute, E., & Wolsey, L. (Eds.) Proceedings of the international workshop on integration of artificial intelligence and operations research techniques in constraint programming for combinatorial optimization problems (CPAIOR 2007). Lecture notes in computer science, (Vol. 4510 pp. 84–98): Springer.Google Scholar
- 19.Hadzic, T., Hooker, J.N., O’Sullivan, B., & Tiedemann, P. (2008). Approximate compilation of constraints into multivalued decision diagrams. In Stuckey, P.J. (Ed.) Principles and practice of constraint programming (CP 2008). Lecture notes in computer science, (Vol. 5202 pp. 448–462): Springer.Google Scholar
- 20.Hadzic, T., Hooker, J.N., & Tiedemann, P. (2008). Propagating separable equalities in an MDD store. In Perron, L., & Trick, M.A. (Eds.) Proceedings of the international workshop on integration of artificial intelligence and operations research techniques in constraint programming for combintaorial optimization problems (CPAIOR 2008). Lecture notes in computer science, (Vol. 5015 pp. 318–322): Springer.Google Scholar
- 21.Hoda, S., Hoeve, W.J.V., & Hooker, J.N. (2010). A systematic approach to MDD-based constraint programming. In Proceedings of the 16th international conference on principles and practices of constraint programming. Lecture notes in computer science, (Vol. 6308 pp. 266–280): Springer.Google Scholar
- 22.Hooker, J.N. (2013). Decision diagrams and dynamic programming. In Gomes, C.P., & Sellmann, M. (Eds.) CPAIOR. Lecture notes in computer science, (Vol. 7874 pp. 94–110): Springer.Google Scholar
- 24.Lokshtanov, D. (2009). New methods in parameterized algorithms and complexity. Ph.D. thesis, University of Bergen.Google Scholar
- 25.Rothvoß, T. (2011). Some 0/1 polytopes need exponential size extended formulations. arXiv:1105.0036.
- 26.Rothvoß, T. (2014). The matching polytope has exponential extension complexity. In Proceedings of the 46th annual ACM symposium on theory of computing (pp. 263–272). New York, NY, USA: STOC ’14, ACM.Google Scholar
- 27.Wegener, I. (2000). Branching programs and binary decision diagrams: theory and applications. SIAM monographs on discrete mathematics and applications. Society for Industrial and Applied Mathematics.Google Scholar