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Fractional 0–1 programming and submodularity

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Abstract

In this note we study multiple-ratio fractional 0–1 programs, a broad class of \(\mathcal {NP}\)-hard combinatorial optimization problems. In particular, under some relatively mild assumptions we provide a complete characterization of the conditions, which ensure that a single-ratio function is submodular. Then we illustrate our theoretical results with the assortment optimization and facility location problems, and discuss practical situations that guarantee submodularity in the considered application settings. In such cases, near-optimal solutions for multiple-ratio fractional 0–1 programs can be found via simple greedy algorithms.

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Acknowledgements

This note is based upon work supported by the National Science Foundation under Grant No. 1818700. The authors would like to thank the Associate Editor and two anonymous referees for their constructive and helpful comments.

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Authors and Affiliations

  1. Industrial and Systems Engineering, University of Southern California, Los Angeles, CA, 90089, USA

    Shaoning Han & Andrés Gómez

  2. Industrial Engineering, University of Pittsburgh, Pittsburgh, PA, 15261, USA

    Oleg A. Prokopyev

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  1. Shaoning Han
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  2. Andrés Gómez
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  3. Oleg A. Prokopyev
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Correspondence to Oleg A. Prokopyev.

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Han, S., Gómez, A. & Prokopyev, O.A. Fractional 0–1 programming and submodularity. J Glob Optim 84, 77–93 (2022). https://doi.org/10.1007/s10898-022-01131-5

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