Computing welfare-Maximizing fair allocations of indivisible goods

Haris Aziz, Xin Huang, Nicholas Mattei, Erel Segal-Halevi

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


We analyze the run-time complexity of computing allocations that are both fair and maximize the utilitarian social welfare, defined as the sum of agents’ utilities. We focus on two tractable fairness concepts: envy-freeness up to one item (EF1) and proportionality up to one item (PROP1). We consider two computational problems: (1) Among the utilitarian-maximal allocations, decide whether there exists one that is also fair; (2) among the fair allocations, compute one that maximizes the utilitarian welfare. We show that both problems are strongly NP-hard when the number of agents is variable, and remain NP-hard for a fixed number of agents greater than two. For the special case of two agents, we find that problem (1) is polynomial-time solvable, while problem (2) remains NP-hard. Finally, with a fixed number of agents, we design pseudopolynomial-time algorithms for both problems. We extend our results to the stronger fairness notions envy-freeness up to any item (EFx) and proportionality up to any item (PROPx).

Original languageEnglish
Pages (from-to)773-784
Number of pages12
JournalEuropean Journal of Operational Research
Issue number2
StatePublished - 1 Jun 2023


  • Assignment
  • Fair division
  • Group decisions and negotiations
  • Indivisible items
  • Utilitarian welfare


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