Asymptotic behavior of the expected optimal value of the multidimensional assignment problem

Authors:
Pavlo A. Krokhmal;Don A. Grundel;Panos M. Pardalos
Affiliations:
University of Iowa, Department of Mechanical and Industrial Engineering, 52242, Iowa City, IA, USA;Air Force Research Lab/Munitions Directorate, 32542, Eglin AFB, FL, USA;University of Florida, Department of Industrial and Systems Engineering, 32611, Gainesville, FL, USA
Venue:
Mathematical Programming: Series A and B
Year:
2007

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Abstract

The Multidimensional Assignment Problem (MAP) is a higher-dimensional version of the Linear Assignment Problem that arises in the areas of data association, target tracking, resource allocation, etc. This paper elucidates the question of asymptotical behavior of the expected optimal value of the large-scale MAP whose assignment costs are independent identically distributed random variables with a prescribed probability distribution. We demonstrate that for a broad class of continuous distributions the limiting value of the expected optimal cost of the MAP is determined by the location of the left endpoint of the support set of the distribution, and construct asymptotical bounds for the expected optimal cost.