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Polyhedral Characterization of Discrete Dynamic Programming

Author

Listed:
  • R. Kipp Martin

    (University of Chicago, Chicago, Illinois)

  • Ronald L. Rardin

    (Purdue University, West Lafayette, Indiana)

  • Brian A. Campbell

    (Purdue University, West Lafayette, Indiana)

Abstract

Many interesting combinatorial problems can be optimized efficiently using recursive computations often termed discrete dynamic programming . In this paper, we develop a paradigm for a general class of such optimizations that yields a polyhedral description for each model in the class. The elementary concept of dynamic programs as shortest path problems in acyclic graphs is generalized to one seeking a least cost solution in a directed hypergraph. Sufficient conditions are then provided for binary integrality of the associated hyperflow problem. Given a polynomially solvable dynamic program, the result is a linear program, in polynomially many variables and constraints, having the solution vectors of the dynamic program as its extreme-point optima. That is, the linear program provides a succinct characterization of the solutions to the underlying optimization problem expressed through an appropriate change of variables. We also discuss projecting this formulation to recover constraints on the original variables and illustrate how some important dynamic programming solvable models fit easily into our paradigm. A classic multiechelon lot sizing problem in production and a host of optimization problems on recursively defined classes of graphs are included.

Suggested Citation

  • R. Kipp Martin & Ronald L. Rardin & Brian A. Campbell, 1990. "Polyhedral Characterization of Discrete Dynamic Programming," Operations Research, INFORMS, vol. 38(1), pages 127-138, February.
    • Handle: RePEc:inm:oropre:v:38:y:1990:i:1:p:127-138
    DOI: 10.1287/opre.38.1.127
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    Citations

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    Cited by:

    1. Yuri Faenza & Volker Kaibel, 2009. "Extended Formulations for Packing and Partitioning Orbitopes," Mathematics of Operations Research, INFORMS, vol. 34(3), pages 686-697, August.
    2. de Lima, Vinícius L. & Alves, Cláudio & Clautiaux, François & Iori, Manuel & Valério de Carvalho, José M., 2022. "Arc flow formulations based on dynamic programming: Theoretical foundations and applications," European Journal of Operational Research, Elsevier, vol. 296(1), pages 3-21.
    3. Christopher Hojny & Tristan Gally & Oliver Habeck & Hendrik Lüthen & Frederic Matter & Marc E. Pfetsch & Andreas Schmitt, 2020. "Knapsack polytopes: a survey," Annals of Operations Research, Springer, vol. 292(1), pages 469-517, September.
    4. Michele Conforti & Gérard Cornuéjols & Giacomo Zambelli, 2013. "Extended formulations in combinatorial optimization," Annals of Operations Research, Springer, vol. 204(1), pages 97-143, April.
    5. Amin Hosseininasab & Willem-Jan van Hoeve, 2021. "Exact Multiple Sequence Alignment by Synchronized Decision Diagrams," INFORMS Journal on Computing, INFORMS, vol. 33(2), pages 721-738, May.
    6. David Bergman & Andre A. Cire, 2018. "Discrete Nonlinear Optimization by State-Space Decompositions," Management Science, INFORMS, vol. 64(10), pages 4700-4720, October.
    7. Steven Harrod, 2011. "Modeling Network Transition Constraints with Hypergraphs," Transportation Science, INFORMS, vol. 45(1), pages 81-97, February.
    8. Christian Tjandraatmadja & Willem-Jan van Hoeve, 2019. "Target Cuts from Relaxed Decision Diagrams," INFORMS Journal on Computing, INFORMS, vol. 31(2), pages 285-301, April.
    9. STEPHAN, Rüdiger, 2010. "An extension of disjunctive programming and its impact for compact tree formulations," LIDAM Discussion Papers CORE 2010045, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    10. François Clautiaux & Ruslan Sadykov & François Vanderbeck & Quentin Viaud, 2019. "Pattern-based diving heuristics for a two-dimensional guillotine cutting-stock problem with leftovers," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 7(3), pages 265-297, September.

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