New Convex Programming Technique for Nash Social Welfare and Scheduling

2026-04-27

AI Summary

This work addresses the approximate optimization of weighted Nash Social Welfare (NSW) maximization and two related variants of unrelated machine scheduling. It introduces a novel convex programming relaxation that, for the first time, reduces the exponentially sized configuration linear program to a polynomial-sized compact formulation while preserving approximation guarantees. Combined with the rounding technique of Feng and Li, the approach is efficiently implementable using standard linear programming solvers, achieving an $e^{1/e} \approx 1.445$ approximation ratio for weighted NSW with only an additive loss of $\ln(1+\varepsilon)$. The method successfully extends to unrelated machine scheduling, matching the current best-known approximation performance. Additionally, the paper provides a constructive proof for the unweighted case and simplifies the analysis of the EF1 fairness gap.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

Abstract

We propose a new convex programming relaxation for the weighted Nash social welfare (NSW) problem that achieves a matching $(e^{1/e}\approx 1.445)$-approximation via the rounding algorithm of Feng and Li. Unlike the exponential-size configuration LP used in prior work, our formulation can be converted into a compact linear program of polynomial size, incurring only an additive loss of $\ln(1+ε)$ in the objective. This allows the program to be solved directly using standard LP solvers, without the ellipsoid method or dual separation oracles. In the unweighted case, we show that our convex program is equivalent to the restricted-spending Fisher market convex program of Cole and Gkatzelis, yielding a constructive proof that its integrality gap is exactly $e^{1/e}$. With a minor modification, our analysis also gives a simple proof of the $e^{1/e}$ EF1 gap for the identical agent setting. Finally, we show that our convex programming technique extends to two unrelated machine scheduling problems, recovering the best-known approximation ratios with simpler analyses.

Problem

• Nash Social Welfare
• convex programming
• scheduling
• approximation algorithm
• Fisher market

Innovation

• convex programming
• Nash social welfare
• linear programming relaxation
• approximation algorithm
• scheduling