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Maximizing Nash Social Welfare in Two-Value Instances: Delineating Tractability

Author

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  • Hannaneh Akrami

    (Max Planck Institute for Informatics, 66123 Saarbrücken, Germany; and Hertz Chair for Algorithms and Optimization, Bonn University, 53113 Bonn, Germany; and Universität des Saarlandes, 66123 Saarbrücken, Germany)

  • Bhaskar Ray Chaudhury

    (Department of Computer Science, University of Illinois, Urbana-Champaign, Urbana, Illinois 61820; and Department of Industrial and Enterprise Systems Engineering, University of Illinois, Urbana-Champaign, Urbana, Illinois 61820)

  • Martin Hoefer

    (Department of Computer Science, RWTH Aachen University, 52062 Aachen, Germany)

  • Kurt Mehlhorn

    (Max Planck Institute for Informatics, 66123 Saarbrücken, Germany; and Universität des Saarlandes, 66123 Saarbrücken, Germany)

  • Marco Schmalhofer

    (Institute for Computer Science, Goethe University Frankfurt, 60629 Frankfurt am Main, Germany)

  • Golnoosh Shahkarami

    (Max Planck Institute for Informatics, 66123 Saarbrücken, Germany; and Saarbrücken Graduate School of Computer Science, Universität des Saarlandes, 66123 Saarbrücken, Germany)

  • Giovanna Varricchio

    (Department of Mathematics and Computer Science, University of Calabria, 87036 Arcavacata, Italy)

  • Quentin Vermande

    (INRIA Centre at Université Côte d’Azur, 06902 Sophia Antipolis, France)

  • Ernest van Wijland

    (IRIF, Université Paris-Cité, 75006 Paris, France)

Abstract

We study the problem of allocating a set of indivisible goods among a set of agents with two-value additive valuations . In this setting, each good is valued either 1 or p / q for some fixed coprime numbers p , q ∈ N such that 1 ≤ q < p . Our goal is to find an allocation that maximizes the Nash social welfare (NSW), that is, the geometric mean of the valuations of the agents. In this work, we give a complete characterization of polynomial-time tractability of NSW maximization that solely depends on the value of q . We start by providing a rather simple polynomial-time algorithm to find a maximum NSW allocation when the valuation functions are integral , that is, q = 1 . We then exploit more involved techniques to get an algorithm that produces a maximum NSW allocation for the half-integral case, that is, q = 2 . Finally, we show it is NP-hard to compute an allocation with maximum NSW whenever q ≥ 3 .

Suggested Citation

  • Hannaneh Akrami & Bhaskar Ray Chaudhury & Martin Hoefer & Kurt Mehlhorn & Marco Schmalhofer & Golnoosh Shahkarami & Giovanna Varricchio & Quentin Vermande & Ernest van Wijland, 2026. "Maximizing Nash Social Welfare in Two-Value Instances: Delineating Tractability," Mathematics of Operations Research, INFORMS, vol. 51(2), pages 853-876, May.
    • Handle: RePEc:inm:ormoor:v:51:y:2026:i:2:p:853-876
    DOI: 10.1287/moor.2023.0204
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