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Exact solution approaches for bilevel assignment problems

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Abstract

We consider the bilevel assignment problem in which each decision maker (i.e., the leader and the follower) has its own objective function and controls a distinct set of edges in a given bipartite graph. The leader acts first by choosing some of its edges. Subsequently, the follower completes the assignment process. The edges selected by the leader and the follower are required to constitute a perfect matching. In this paper we propose an exact solution approach, which is based on a branch-and-bound framework and exploits structural properties of the assignment problem. Extensive computational experiments with linear sum and linear bottleneck objective functions are conducted to demonstrate the performance of the developed methods. While the considered problem is known to be NP-hard in general, we also describe some restricted cases that can be solved in polynomial time.

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Acknowledgments

The authors would like to thank the Associate Editor and the anonymous reviewers for their constructive comments that helped us to greatly improve the quality of the paper. This material is based upon work supported by the U.S. Air Force Research Laboratory (AFRL) Mathematical Modeling and Optimization Institute and the U.S. Air Force Office of Scientific Research (AFOSR). The research of the second author was also supported by the U.S. Air Force Summer Faculty Fellowship and by AFRL/RW under agreement number FA8651-12-2-0008. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of AFRL/RW or the U.S. Government.

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Authors and Affiliations

  1. Department of Industrial Engineering, University of Pittsburgh, 1048 Benedum Hall, Pittsburgh, PA, 15261, USA

    Behdad Beheshti & Oleg A. Prokopyev

  2. AFRL Munitions Directorate, Building 13, Eglin AFB, FL, 32542, USA

    Eduardo L. Pasiliao

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  1. Behdad Beheshti
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  2. Oleg A. Prokopyev
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  3. Eduardo L. Pasiliao
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Correspondence to Oleg A. Prokopyev.

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Beheshti, B., Prokopyev, O.A. & Pasiliao, E.L. Exact solution approaches for bilevel assignment problems. Comput Optim Appl 64, 215–242 (2016). https://doi.org/10.1007/s10589-015-9799-4

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