Solving the Multidimensional Assignment Problem by a Cross-Entropy method

Authors:
Duc Manh Nguyen;Hoai An Le Thi;Tao Pham Dinh
Affiliations:
LMI, National Institute for Applied Sciences (INSA) --- Rouen, Saint-Étienne-du-Rouvray, France 76801;Laboratory of Theoretical and Applied Computer Science (LITA), University of Lorraine, Metz Cedex, France 57045;LMI, National Institute for Applied Sciences (INSA) --- Rouen, Saint-Étienne-du-Rouvray, France 76801
Venue:
Journal of Combinatorial Optimization
Year:
2014

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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.