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Constant Approximation for Weighted Nash Social Welfare with Submodular Valuations

Yuda Feng, Yang Hu, Shi Li, Ruilong Zhang

TL;DR

The paper addresses maximizing weighted Nash Social Welfare under submodular valuations, a problem previously approximated only by O(n w_{\max}). It introduces a Configuration LP-based framework combined with a large-item/small-item decomposition and a Li-style rounding, ensuring exactly one fractional large item per agent and negatively correlated small-item allocations through iterative rounding on a bipartite multi-graph. The authors prove concentration bounds for the rounded allocations and relate the LP objective to the final NSW via careful per-agent analysis, yielding a constant-factor approximation (~233+\ε). This result resolves a longstanding open question and significantly advances the understanding of fair and efficient resource allocation for submodular utilities, with implications for related scheduling and economic settings.

Abstract

We study the problem of assigning items to agents so as to maximize the \emph{weighted} Nash Social Welfare (NSW) under submodular valuations. The best-known result for the problem is an $O(nw_{\max})$-approximation due to Garg, Husic, Li, Végh, and Vondrák~[STOC 2023], where $w_{\max}$ is the maximum weight over all agents. Obtaining a constant approximation algorithm is an open problem in the field that has recently attracted considerable attention. We give the first such algorithm for the problem, thus solving the open problem in the affirmative. Our algorithm is based on the natural Configuration LP for the problem, which was introduced recently by Feng and Li~[ICALP 2024] for the additive valuation case. Our rounding algorithm is similar to that of Li~[SODA 2025] developed for the unrelated machine scheduling problem to minimize weighted completion time. Roughly speaking, we designate the largest item in each configuration as a large item and the remaining items as small items. So, every agent gets precisely 1 fractional large item in the configuration LP solution. With the rounding algorithm in Li~[SODA 2025], we can ensure that in the obtained solution, every agent gets precisely 1 large item, and the assignments of small items are negatively correlated.

Constant Approximation for Weighted Nash Social Welfare with Submodular Valuations

TL;DR

The paper addresses maximizing weighted Nash Social Welfare under submodular valuations, a problem previously approximated only by O(n w_{\max}). It introduces a Configuration LP-based framework combined with a large-item/small-item decomposition and a Li-style rounding, ensuring exactly one fractional large item per agent and negatively correlated small-item allocations through iterative rounding on a bipartite multi-graph. The authors prove concentration bounds for the rounded allocations and relate the LP objective to the final NSW via careful per-agent analysis, yielding a constant-factor approximation (~233+\ε). This result resolves a longstanding open question and significantly advances the understanding of fair and efficient resource allocation for submodular utilities, with implications for related scheduling and economic settings.

Abstract

We study the problem of assigning items to agents so as to maximize the \emph{weighted} Nash Social Welfare (NSW) under submodular valuations. The best-known result for the problem is an -approximation due to Garg, Husic, Li, Végh, and Vondrák~[STOC 2023], where is the maximum weight over all agents. Obtaining a constant approximation algorithm is an open problem in the field that has recently attracted considerable attention. We give the first such algorithm for the problem, thus solving the open problem in the affirmative. Our algorithm is based on the natural Configuration LP for the problem, which was introduced recently by Feng and Li~[ICALP 2024] for the additive valuation case. Our rounding algorithm is similar to that of Li~[SODA 2025] developed for the unrelated machine scheduling problem to minimize weighted completion time. Roughly speaking, we designate the largest item in each configuration as a large item and the remaining items as small items. So, every agent gets precisely 1 fractional large item in the configuration LP solution. With the rounding algorithm in Li~[SODA 2025], we can ensure that in the obtained solution, every agent gets precisely 1 large item, and the assignments of small items are negatively correlated.

Paper Structure

This paper contains 22 sections, 15 theorems, 29 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Result.
  3. Overview of Our Techniques
  4. Organization.
  5. Preliminaries
  6. Truncations of Submodular Functions
  7. Extensions of Submodular Functions
  8. Concentration Bound for Submodular Functions in Pipage Rounding
  9. Iterative Rounding for Weighted Submodular Nash Social Welfare using Configuration LP
  10. LP Formulation
  11. Construction of Bipartite Multi-Graph $G = (N \cup M, E)$ with Marked and Unmarked Edges
  12. Rounding $\mathbf{x}$ into an Integral Solution
  13. Analysis of the Approximation Ratio
  14. Input and Output Distributions
  15. Notations and Histogram for Input Distribution.
  16. ...and 7 more sections

Key Result

Theorem 1.1

For any $\epsilon>0$, there is a randomized $(233+\epsilon)$-approximation algorithm for the weighted Nash social welfare problem with submodular valuations, with running time polynomial in the size of the input and $\frac{1}{\epsilon}$.

Figures (2)

  • Figure 1: Illustration for the constructed bipartite multi-graph. There are two agents and three items, and the agents' preference for items is shown in the figure. Suppose that, after solving \ref{['Conf-LP']}, we obtain $y^*_{1,S_1}=0.3,y^*_{1,S_2}=0.7$, $y^*_{2,S_3}=0.3,y^*_{2,S_1}=0.4,y^*_{2,S_4}=0.3$. By agents' preference, we know that $\largeitems[1]=\set{j_1,j_2}$ and $\largeitems[2]=\set{j_1,j_3}$, they are marked by gray cycle. The small item set is $\smallitems[1]=\set{j_3},\smallitems[2]=\set{j_1,j_2}$. Then, we can create a bipartite multi-graph as stated above. In the right part of the figure, we use a solid line to represent the marked edges (edges between agents and large items) and a dashed line to represent the unmarked edges (edges between agents and small items). The example shows two edges between agent $a_2$ and item $j_1$; thus, the bipartite graph can be a multi-graph. Verifying the two properties of the constructed vector $\mathbf{x}^*$ is also easy.
  • Figure 2: Illustration for the $\pi(\cdot)$ function and $C_t$, which are shown in the subfigure (i) and (ii), respectively. For simplicity, the figure considers the case when the valuation function $v$ is additive. In the subfigure (i), for each set $S\subseteq M$ with $y^*_S>0$, we have a rectangle. We sort these rectangles in non-increasing order according to the value of the largest item in each set. In each rectangle, the gray part is the size of the largest item, and the remaining part is the small item part. In the subfigure (ii), for each possible algorithm's output of small items, we have a rectangle. We also sort these rectangles in non-increasing order according to the value of each set.

Theorems & Definitions (35)

  • Theorem 1.1
  • Definition 2.1: Monotone Submodular Functions
  • Definition 2.2: Truncated Function
  • Lemma 2.3
  • proof
  • Definition 2.4: Multilinear Extension
  • Definition 2.5: Concave Extension
  • Lemma 2.6: vondrak2007submodularity
  • Theorem 2.7
  • Lemma 3.1
  • ...and 25 more