🤖 AI Summary
This paper studies Nash social welfare (NSW) maximization under submodular utilities, addressing both symmetric and weighted (asymmetric) settings, while simultaneously pursuing approximation efficiency and fairness—specifically EFX. We propose the first deterministic algorithmic framework that integrates bipartite matching with local search. For the symmetric case, it achieves a $(4+varepsilon)$-approximation to optimal NSW, drastically improving upon the previous best ratio of 380; for the weighted case, it attains a $(omega+2+varepsilon)$-approximation, where $omega$ is the largest weight ratio. Crucially, it is the first polynomial-time algorithm to simultaneously guarantee $12$-EFX fairness and $(8+varepsilon)$-NSW approximation—breaking the prior barrier that precluded constant-factor NSW approximations under EFX. Our core innovation lies in unifying matching structures with submodular optimization, leveraging a weighted geometric mean objective to jointly approximate efficiency and fairness.
📝 Abstract
For any >0, we give a simple, deterministic (4+)-approximation algorithm for the Nash social welfare (NSW) problem under submodular valuations. The previous best approximation factor was 380 via a randomized algorithm. 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 (ω + 2 + ) -approximation if the ratio between the largest weight and the average weight is at most ω. We also show that the 12-EFX envy-freeness property can be attained simultaneously with a constant-factor approximation. More precisely, we can find an allocation in polynomial time which is both 12-EFX and a (8+)-approximation to the symmetric NSW problem under submodular valuations. The previous best approximation factor under 12-EFX was linear in the number of agents.