New Convex Programming Technique for Nash Social Welfare and Scheduling

TL;DR

This paper introduces a new convex programming relaxation for the Nash social welfare problem that is compact, efficient, and achieves a near-optimal approximation ratio, extending to scheduling problems.

Contribution

A novel convex relaxation for NSW that is polynomial-sized, matches the best approximation ratio, and simplifies solving and analysis compared to prior exponential formulations.

Findings

Achieves a $(e^{1/e})$-approximation for weighted NSW.

Provides a polynomial-size LP formulation that can be solved with standard solvers.

Extends the convex programming approach to machine scheduling problems with optimal ratios.

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 $\ln(1+\epsilon)$ 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.