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Computational complexity of necessary envy-freeness

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  • Aziz, Haris
  • Schlotter, Ildikó
  • Walsh, Toby

Abstract

We consider the fundamental problem of fairly allocating indivisible items when agents have strict ordinal preferences over individual items. We focus on the well-studied fairness criterion of necessary envy-freeness. For a constant number of agents, the computational complexity of the deciding whether there exists an allocation that satisfies necessary envy-freeness has been open for several years. We settle this question by showing that the problem is NP-complete even for three agents. Considering that the problem is polynomial-time solvable for the case of two agents, we provide a clear understanding of the complexity of the problem with respect to the number of agents.

Suggested Citation

  • Aziz, Haris & Schlotter, Ildikó & Walsh, Toby, 2024. "Computational complexity of necessary envy-freeness," Mathematical Social Sciences, Elsevier, vol. 127(C), pages 86-98.
    • Handle: RePEc:eee:matsoc:v:127:y:2024:i:c:p:86-98
    DOI: 10.1016/j.mathsocsci.2023.08.002
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    Keywords

    Fair division; Envy-freeness;

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