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On the number of local minima for the multidimensional assignment problem

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Abstract

The Multidimensional Assignment Problem (MAP) is an NP-hard combinatorial optimization problem occurring in many applications, such as data association, target tracking, and resource planning. As many solution approaches to this problem rely, at least partly, on local neighborhood search algorithms, the number of local minima affects solution difficulty for these algorithms. This paper investigates the expected number of local minima in randomly generated instances of the MAP. Lower and upper bounds are developed for the expected number of local minima, E[M], in an MAP with iid standard normal coefficients. In a special case of the MAP, a closed-form expression for E[M] is obtained when costs are iid continuous random variables. These results imply that the expected number of local minima is exponential in the number of dimensions of the MAP. Our numerical experiments indicate that larger numbers of local minima have a statistically significant negative effect on the quality of solutions produced by several heuristic algorithms that involve local neighborhood search.

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Authors and Affiliations

  1. Analysis Division, Air Armament Center, Eglin AFB, FL, 32542, Eglin, USA

    Don A. Grundel

  2. Department of Mechanical and Industrial Engineering, The University of Iowa, 2403 Seamans Center for the Engineering Arts and Sciences, Iowa City, IA, 55242, USA

    Pavlo A. Krokhmal

  3. School of Industrial Engineering and Management, Oklahoma State University, 322 Engineering North, Stillwater, OK, 74078, USA

    Carlos A. S. Oliveira

  4. Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, 303 Weil Hall, Gainesville, FL, 32611, USA

    Panos M. Pardalos

Authors
  1. Don A. Grundel
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  2. Pavlo A. Krokhmal
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  3. Carlos A. S. Oliveira
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  4. Panos M. Pardalos
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Corresponding author

Correspondence to Pavlo A. Krokhmal.

Additional information

Partially supported by the NSF grant DMI-0457473.

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Grundel, D.A., Krokhmal, P.A., Oliveira, C.A.S. et al. On the number of local minima for the multidimensional assignment problem. J Comb Optim 13, 1–18 (2007). https://doi.org/10.1007/s10878-006-9009-5

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  • DOI: https://doi.org/10.1007/s10878-006-9009-5

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