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An LP-based characterization of solvable QAP instances with chess-board and graded structures

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Abstract

The quadratic assignment problem (QAP) is perhaps the most widely studied nonlinear combinatorial optimization problem. It has many applications in various fields, yet has proven to be extremely difficult to solve. This difficulty has motivated researchers to identify special objective function structures that permit an optimal solution to be found efficiently. Previous work has shown that certain such structures can be explained in terms of a mixed 0–1 linear reformulation of the QAP known as the level-1 reformulation–linearization-technique (RLT) form. Specifically, the objective function structures were shown to ensure that a binary optimal extreme point solution exists to the continuous relaxation. This paper extends that work by considering classes of solvable cases in which the objective function coefficients have special chess-board and graded structures, and similarly characterizing them in terms of the level-1 RLT form. As part of this characterization, we develop a new relaxed version of the level-1 RLT form, the structure of which can be readily exploited to study the special instances under consideration.

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Funding

Author L. Waddell’s work was supported by Department of Navy award N00014-20-1-2072 issued by the Office of Naval Research. Author T. Liu’s work was supported by Bucknell University’s Program for Undergraduate Research.

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Authors and Affiliations

  1. Bucknell University, Lewisburg, PA, 17837, USA

    Lucas A. Waddell, Tianzhu Liu & Swarup Dhar

  2. Francis Marion University, Florence, SC, 29502, USA

    Jerry L. Phillips

Authors
  1. Lucas A. Waddell
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  2. Jerry L. Phillips
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  3. Tianzhu Liu
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  4. Swarup Dhar
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Contributions

This study was conceived by authors L. Waddell and J. Phillips. All authors contributed to the development and proofs of the theoretical results contained in this paper. All authors contributed to, read, and approved the final manuscript.

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Correspondence to Lucas A. Waddell.

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The work relates to Department of Navy award N00014-20-1-2072 issued by the Office of Naval Research. The United States Government has a royalty-free license throughout the world in all copyrightable material contained herein. The authors would also like to thank Bucknell University’s Program for Undergraduate Research for its support.

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Waddell, L.A., Phillips, J.L., Liu, T. et al. An LP-based characterization of solvable QAP instances with chess-board and graded structures. J Comb Optim 45, 114 (2023). https://doi.org/10.1007/s10878-023-01044-3

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