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Graph partitions for the multidimensional assignment problem

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Abstract

In this paper, we consider two decomposition schemes for the graph theoretical description of the axial Multidimensional Assignment Problem (MAP). The problem is defined as finding n disjoint cliques of size m with minimum total cost in K m×n , which is an m-partite graph with n elements per dimension. Even though the 2-dimensional assignment problem is solvable in polynomial time, extending the problem to include n≥3 dimensions renders it \(\mathcal{NP}\)-hard. We propose two novel decomposition schemes for partitioning a MAP into disjoint subproblems, that can then be recombined to provide both upper and lower bounds to the original problem. For each of the partitioning schemes, we investigate and compare the efficiency of distinct exact and heuristic methodologies, namely augmentation and partitioning. Computational results for the methods, along with a hybrid one that consists of both partitioning schemes, are presented to depict the success of our approaches on large-scale instances.

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Acknowledgement

This research was supported by the AFRL Mathematical Modeling and Optimization Institute.

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Authors and Affiliations

  1. Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA

    Chrysafis Vogiatzis & Panos M. Pardalos

  2. Munitions Directorate, Air Force Research Laboratory, Eglin AFB, FL, 32542, USA

    Eduardo L. Pasiliao

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  1. Chrysafis Vogiatzis
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  2. Eduardo L. Pasiliao
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  3. Panos M. Pardalos
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Correspondence to Chrysafis Vogiatzis.

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Vogiatzis, C., Pasiliao, E.L. & Pardalos, P.M. Graph partitions for the multidimensional assignment problem. Comput Optim Appl 58, 205–224 (2014). https://doi.org/10.1007/s10589-013-9619-7

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