Approximating Nash Social Welfare by Matching and Local Search
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.