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Dynamic Sparsification for Quadratic Assignment Problems

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Book cover Mathematical Optimization Theory and Operations Research (MOTOR 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11548))

Abstract

We present a framework for optimizing sparse quadratic assignment problems. We propose an iterative algorithm that dynamically generates the quadratic part of the assignment problem and, thus, solves a sparsified linearization of the original problem in every iteration. This procedure results in a hierarchy of lower bounds and, in addition, provides heuristic primal solutions in every iteration. This framework was motivated by the task of the French government to design the French keyboard standard, which included solving sparse quadratic assignment problems with over 100 special characters; a size where many commonly used approaches fail. The design of a new standard often involves conflicting opinions of multiple stakeholders in a committee. Hence, there is no agreement on a single well-defined objective function that can be used for an extensive one-shot optimization. Instead, the process is highly interactive and demands rapid prototyping, e.g., quick primal solutions, on-the-fly evaluation of manual changes, and prompt assessments of solution quality. Particularly concerning the latter aspect, our algorithm is able to provide high-quality lower bounds for these problems in several minutes.

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Notes

  1. 1.

    https://normalisation.afnor.org/actualites/faq-clavier-francais/ – retr. 2019-04-03.

  2. 2.

    We give no polynomial time guarantee. The existence of a PTAS would imply P = NP.

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Authors and Affiliations

  1. Max Planck Institute for Informatics, Saarbrücken, Germany

    Maximilian John & Andreas Karrenbauer

Authors
  1. Maximilian John
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  2. Andreas Karrenbauer
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Correspondence to Maximilian John .

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Editors and Affiliations

  1. Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Yury Kochetov

  3. University of Florida, Gainesville, FL, USA

    Panos Pardalos

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John, M., Karrenbauer, A. (2019). Dynamic Sparsification for Quadratic Assignment Problems. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2019. Lecture Notes in Computer Science(), vol 11548. Springer, Cham. https://doi.org/10.1007/978-3-030-22629-9_17

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  • DOI: https://doi.org/10.1007/978-3-030-22629-9_17

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