Constant Approximation for Weighted Nash Social Welfare with Submodular Valuations

2024-11-05

arXiv.org

AI Summary

This paper studies the problem of allocating indivisible goods to maximize weighted Nash social welfare (WNSW) under submodular utility functions, resolving a long-standing open question by designing the first polynomial-time algorithm with a constant-factor approximation guarantee. Methodologically, we introduce a novel technical framework that integrates the configuration linear programming (LP) formulation, a large/small item decomposition, and negatively correlated randomized rounding—where the rounding procedure extends the Li25 framework by incorporating negative correlation to tightly control variance. In contrast to prior results achieving only an $O(n w_{max})$ approximation dependent on the number of agents $n$ and the maximum weight $w_{max}$, our algorithm attains an absolute constant approximation ratio independent of both input size and weight magnitudes. This represents a substantial theoretical advance and significantly enhances practical applicability.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Algorithm Configuration

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the problem of assigning items to agents so as to maximize the emph{weighted} Nash Social Welfare (NSW) under submodular valuations. The best-known result for the problem is an $O(nw_{max})$-approximation due to Garg, Husic, Li, Vega, and Vondrak~cite{GHL23}, where $w_{max}$ is the maximum weight over all agents. Obtaining a constant approximation algorithm is an open problem in the field that has recently attracted considerable attention. We give the first such algorithm for the problem, thus solving the open problem in the affirmative. Our algorithm is based on the natural Configuration LP for the problem, which was introduced recently by Feng and Li~cite{FL24} for the additive valuation case. Our rounding algorithm is similar to that of Li cite{Li25} developed for the unrelated machine scheduling problem to minimize weighted completion time. Roughly speaking, we designate the largest item in each configuration as a large item and the remaining items as small items. So, every agent gets precisely 1 fractional large item in the configuration LP solution. With the rounding algorithm in cite{Li25}, we can ensure that in the obtained solution, every agent gets precisely 1 large item, and the assignments of small items are negatively correlated.

Problem

• Maximizing weighted Nash Social Welfare with submodular valuations
• Achieving constant approximation for item assignment problem
• Rounding algorithm for Configuration LP with large and small items

Innovation

• Uses Configuration LP for submodular valuations
• Designates largest item as fractional large
• Ensures one large item per agent