Approximating the nash social welfare with indivisible items

Richard Cole, Vasilis Gkatzelis

Research output: Contribution to journalArticlepeer-review


We study the problem of allocating a set of indivisible items among agents with additive valuations, with the goal of maximizing the geometric mean of the agents' valuations, i.e., the Nash social welfare. This problem is known to be NP-hard, and our main result is the first efficient constant-factor approximation algorithm for this objective. We first observe that the integrality gap of the natural fractional relaxation is exponential, so we propose a different fractional allocation which implies a tighter upper bound and, after appropriate rounding, yields a good integral allocation. An interesting contribution of this work is the fractional allocation that we use. The relaxation of our problem can be solved efficiently using the Eisenberg--Gale program, whose optimal solution can be interpreted as a market equilibrium with the dual variables playing the role of item prices. Using this market-based interpretation, we define an alternative equilibrium allocation where the amount of spending that can go into any given item is bounded, thus keeping the highly priced items under-allocated and forcing the agents to spend on lower priced items. The resulting equilibrium prices reveal more information regarding how to assign items so as to obtain a good integral allocation.

Original languageEnglish (US)
Pages (from-to)1211-1236
Number of pages26
JournalSIAM Journal on Computing
Issue number3
StatePublished - 2018


  • Approximation algorithms
  • Competitive equilibrium from equal incomes
  • Fair division
  • Nash social welfare

ASJC Scopus subject areas

  • General Computer Science
  • General Mathematics


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