Approximating Nash Social Welfare by Matching and Local Search
For researchers in fair division and algorithmic game theory, this provides improved approximation guarantees for a fundamental welfare objective under general valuations.
The paper presents a deterministic $(4+\\varepsilon)$-approximation algorithm for Nash social welfare under submodular valuations, and extends to asymmetric NSW with an $e(\\omega+2+\\varepsilon)$-approximation. It also achieves a $(8+\\varepsilon)$-approximation simultaneously with $1/2$-EFX envy-freeness.
For any $\varepsilon>0$, we give a simple, deterministic $(4+\varepsilon)$-approximation algorithm for the Nash social welfare (NSW) problem under submodular valuations. We also consider the asymmetric variant of the problem, where the objective is to maximize the weighted geometric mean of agents' valuations, and give an $e (Ï+ 2 + \varepsilon)$-approximation if the ratio between the largest weight and the average weight is at most $Ï$. We also show that the $1/2$-EFX envy-freeness property can be attained simultaneously with a constant-factor approximation. More precisely, we can find an allocation in polynomial time that is both $1/2$-EFX and a $(8+\varepsilon)$-approximation to the symmetric NSW problem under submodular valuations.