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Solving the Multidimensional Assignment Problem by a Cross-Entropy method

Author

Listed:
  • Duc Manh Nguyen

    (National Institute for Applied Sciences (INSA) – Rouen)

  • Hoai An Le Thi

    (University of Lorraine)

  • Tao Pham Dinh

    (National Institute for Applied Sciences (INSA) – Rouen)

Abstract

The Multidimensional Assignment Problem (MAP) is a higher dimensional version of the linear assignment problem, where we find tuples of elements from given sets, such that the total cost of the tuples is minimal. The MAP has many recognized applications such as data association, target tracking, and resource planning. While the linear assignment problem is solvable in polynomial time, the MAP is NP-hard. In this work, we develop a new approach based on the Cross-Entropy (CE) methods for solving the MAP. Exploiting the special structure of the MAP, we propose an appropriate family of discrete distributions on the feasible set of the MAP that allow us to design an efficient and scalable CE algorithm. The efficiency and scalability of our method are proved via several tests on large-scale problems with up to 5 dimensions and 20 elements in each dimension, which is equivalent to a 0–1 linear program with 3.2 millions binary variables and 100 constraints.

Suggested Citation

  • Duc Manh Nguyen & Hoai An Le Thi & Tao Pham Dinh, 2014. "Solving the Multidimensional Assignment Problem by a Cross-Entropy method," Journal of Combinatorial Optimization, Springer, vol. 27(4), pages 808-823, May.
    • Handle: RePEc:spr:jcomop:v:27:y:2014:i:4:d:10.1007_s10878-012-9554-z
    DOI: 10.1007/s10878-012-9554-z
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    References listed on IDEAS

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    1. Renata M. Aiex & Mauricio G. C. Resende & Panos M. Pardalos & Gerardo Toraldo, 2005. "GRASP with Path Relinking for Three-Index Assignment," INFORMS Journal on Computing, INFORMS, vol. 17(2), pages 224-247, May.
    2. William P. Pierskalla, 1968. "Letter to the Editor—The Multidimensional Assignment Problem," Operations Research, INFORMS, vol. 16(2), pages 422-431, April.
    3. H-J Bandelt & A Maas & F C R Spieksma, 2004. "Local search heuristics for multi-index assignment problems with decomposable costs," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 55(7), pages 694-704, July.
    4. Eduardo L. Pasiliao, 2010. "Local Neighborhoods for the Multidimensional Assignment Problem," Springer Optimization and Its Applications, in: Michael J. Hirsch & Panos M. Pardalos & Robert Murphey (ed.), Dynamics of Information Systems, chapter 0, pages 353-371, Springer.
    5. Rubinstein, Reuven Y., 1997. "Optimization of computer simulation models with rare events," European Journal of Operational Research, Elsevier, vol. 99(1), pages 89-112, May.
    6. Egon Balas & Matthew J. Saltzman, 1991. "An Algorithm for the Three-Index Assignment Problem," Operations Research, INFORMS, vol. 39(1), pages 150-161, February.
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