Approximation Algorithms for the Weighted Nash Social Welfare via Convex and Non-Convex Programs
Adam Brown, Aditi Laddha, Madhusudhan Reddy Pittu, Mohit Singh
TL;DR
This work studies the weighted Nash Social Welfare problem with indivisible items and additive valuations, introducing a pair of mathematical relaxations—one convex (CVX) and one non-convex (NCVX)—to enable efficient approximation. The authors prove a polynomial-time algorithm that achieves NSW(σ) ≥ OPT − 2 log 2 − 1/(2e) − 2 D_{KL}(w || u), i.e., a weight-distribution-aware approximation that tightens when the weight vector is close to uniform. The key technical contribution is exploiting two linked relaxations: solving the convex program to obtain a structure that can be sparsified into an acyclic forest, and then rounding via the non-convex relaxation with guarantees tied to the KL-divergence between the weight distribution and uniform. This framework unifies and extends unweighted NSW relaxations, yielding meaningful guarantees that improve under certain weight configurations and suggesting directions for constant-factor results and extensions to submodular valuations. Open questions include whether the KL term is essential for constant-factor bounds and how to generalize the methods to broader valuation classes.
Abstract
In an instance of the weighted Nash Social Welfare problem, we are given a set of $m$ indivisible items, $\mathscr{G}$, and $n$ agents, $\mathscr{A}$, where each agent $i \in \mathscr{A}$ has a valuation $v_{ij}\geq 0$ for each item $j\in \mathscr{G}$. In addition, every agent $i$ has a non-negative weight $w_i$ such that the weights collectively sum up to $1$. The goal is to find an assignment $σ:\mathscr{G}\rightarrow \mathscr{A}$ that maximizes $\prod_{i\in \mathscr{A}} \left(\sum_{j\in σ^{-1}(i)} v_{ij}\right)^{w_i}$, the product of the weighted valuations of the players. When all the weights equal $\frac1n$, the problem reduces to the classical Nash Social Welfare problem, which has recently received much attention. In this work, we present a $5\cdot\exp\left(2\cdot D_{\text{KL}}(\mathbf{w}\, ||\, \frac{\vec{\mathbf{1}}}{n})\right) = 5\cdot\exp\left(2\log{n} + 2\sum_{i=1}^n w_i \log{w_i}\right)$-approximation algorithm for the weighted Nash Social Welfare problem, where $D_{\text{KL}}(\mathbf{w}\, ||\, \frac{\vec{\mathbf{1}}}{n})$ denotes the KL-divergence between the distribution induced by $\mathbf{w}$ and the uniform distribution on $[n]$. We show a novel connection between the convex programming relaxations for the unweighted variant of Nash Social Welfare presented in \cite{cole2017convex, anari2017nash}, and generalize the programs to two different mathematical programs for the weighted case. The first program is convex and is necessary for computational efficiency, while the second program is a non-convex relaxation that can be rounded efficiently. The approximation factor derives from the difference in the objective values of the convex and non-convex relaxation.