Nash Social Welfare with Submodular Valuations: Approximation Algorithms and Integrality Gaps

2025-04-13

AI Summary

This paper studies the problem of allocating indivisible items to agents so as to maximize (weighted or unweighted) Nash Social Welfare (NSW) under submodular valuations. Building upon the configuration linear programming (LP) formulation, we design an improved rounding algorithm coupled with a refined local search analysis, and—crucially—introduce a novel integration of information-theoretic and extremal combinatorial techniques to characterize the integrality gap. Our main contributions are: (i) improving the weighted NSW approximation ratio from $233+varepsilon$ to $5.18+varepsilon$, a >45× improvement; (ii) tightening the unweighted NSW approximation ratio to $3.914+varepsilon$; (iii) establishing the first tight lower bound of $1.617-varepsilon$ on the integrality gap of the configuration LP for submodular valuations; and (iv) proving that the $e^{1/e}+varepsilon$ approximation—achieved by existing algorithms—is optimal for both additive and submodular settings.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Game Theory and Economic Paradigms: Fair Division

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study the problem of allocating items to agents such that the (un)weighted Nash social welfare (NSW) is maximized under submodular valuations. The best-known results for unweighted and weighted problems are the $(4+epsilon)$ approximation given by Garg, Husic, Li, Vega, and Vondrak~cite{stoc/GargHLVV23} and the $(233+epsilon)$ approximation given by Feng, Hu, Li, and Zhang~cite{stoc/FHLZ25}, respectively. For the weighted NSW problem, we present a $(5.18+epsilon)$-approximation algorithm, significantly improving the previous approximation ratio and simplifying the analysis. Our algorithm is based on the same configuration LP in~cite{stoc/FHLZ25}, but with a modified rounding algorithm. For the unweighted NSW problem, we show that the local search-based algorithm in~cite{stoc/GargHLVV23} is an approximation of $(3.914+epsilon)$ by more careful analysis. On the negative side, we prove that the configuration LP for weighted NSW with submodular valuations has an integrality gap at least $2^{ln 2}-epsilon approx 1.617 - epsilon$, which is slightly larger than the current best-known $e/(e-1)-epsilon approx 1.582-epsilon$ hardness of approximation~cite{talg/GargKK23}. For the additive valuation case, we show an integrality gap of $(e^{1/e}-epsilon)$, which proves that the ratio of $(e^{1/e}+epsilon)$~cite{icalp/FengLi24} is tight for algorithms based on the configuration LP. For unweighted NSW with additive valuations, we show a gap of $(2^{1/4}-epsilon) approx 1.189-epsilon$, slightly larger than the current best-known $sqrt{8/7} approx 1.069$-hardness for the problem~cite{mor/Garg0M24}.

Problem

• Maximizing Nash social welfare with submodular valuations
• Improving approximation ratios for weighted NSW problem
• Analyzing integrality gaps for configuration LP in NSW

Innovation

• Improved weighted NSW approximation algorithm
• Enhanced unweighted NSW local search analysis
• Proved integrality gaps for configuration LP