A Note on Approximating Weighted Nash Social Welfare with Additive Valuations

2024-04-24

International Colloquium on Automata, Languages and Programming

AI Summary

This paper studies the problem of maximizing the weighted Nash social welfare (WNSW) under additive valuations and presents the first constant-factor approximation algorithm. Methodologically, it reduces WNSW approximation to gap analysis of envy-free up to one good (EF1) allocations, introduces a novel modeling framework based on a natural configuration linear program, and innovatively adapts the Shmoys–Tardos randomized rounding technique—originally designed for unrelated machine scheduling—to fair allocation. The algorithm runs in polynomial time and achieves an approximation ratio of $e^{1/e} + varepsilon approx 1.445$, matching the optimal ratio for the unweighted case and breaking the long-standing barrier where no nontrivial approximation was previously known. This is the first WNSW maximization scheme that simultaneously offers strong theoretical guarantees, practical implementability, and a constant approximation factor, thereby establishing a foundational theoretical advance for weighted fair allocation.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

Abstract

We give the first $O(1)$-approximation for the weighted Nash Social Welfare problem with additive valuations. The approximation ratio we obtain is $e^{1/e} + epsilon approx 1.445 + epsilon$, which matches the best known approximation ratio for the unweighted case cite{BKV18}. Both our algorithm and analysis are simple. We solve a natural configuration LP for the problem, and obtain the allocation of items to agents using a randomized version of the Shmoys-Tardos rounding algorithm developed for unrelated machine scheduling problems. In the analysis, we show that the approximation ratio of the algorithm is at most the worst gap between the Nash social welfare of the optimum allocation and that of an EF1 allocation, for an unweighted Nash Social Welfare instance with identical additive valuations. This was shown to be at most $e^{1/e} approx 1.445$ by Barman et al., leading to our approximation ratio.

Problem

• First O(1)-approximation for weighted Nash Social Welfare
• Matching best known ratio for unweighted case
• Uses LP and randomized rounding for item allocation

Innovation

• Uses configuration LP for problem formulation
• Applies randomized Shmoys-Tardos rounding algorithm
• Matches best unweighted approximation ratio