Approximating Nash Social Welfare in 2-Valued Instances

07/19/2021 ∙ by Martin Hoefer, et al. ∙ 0 ∙

We consider the problem of maximizing the Nash social welfare when allocating a set 𝒢 of goods to a set 𝒩 of agents. We study instances, in which all agents have 2-valued additive valuations. In such an instance, the value of every agent i ∈𝒩 for every item j ∈𝒢 is v_ij∈{p,q}, for p ≤ q ∈ℕ. We show that an optimal allocation can be computed in polynomial time if p divides q. When p does not divide q, we show an approximation ratio of at most 1.033. It strictly decreases with the denominator of the irreducible fraction that equals p/q. Finally, we prove an APX-hardness result for the problem with a lower bound on the ratio of 1.000015.

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