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Solving the Multidimensional Assignment Problem by a Cross-Entropy method

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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.

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Authors and Affiliations

  1. LMI, National Institute for Applied Sciences (INSA) – Rouen, BP08-Avenue de l’Université, 76801, Saint-Étienne-du-Rouvray, France

    Duc Manh Nguyen & Tao Pham Dinh

  2. Laboratory of Theoretical and Applied Computer Science (LITA), University of Lorraine, Ile du Saulcy, 57045, Metz Cedex, France

    Hoai An Le Thi

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  1. Duc Manh Nguyen
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  2. Hoai An Le Thi
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  3. Tao Pham Dinh
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Correspondence to Hoai An Le Thi.

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Nguyen, D.M., Le Thi, H.A. & Pham Dinh, T. Solving the Multidimensional Assignment Problem by a Cross-Entropy method. J Comb Optim 27, 808–823 (2014). https://doi.org/10.1007/s10878-012-9554-z

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