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

TL;DR

The paper tackles maximizing weighted Nash Social Welfare for agents with submodular valuations, presenting a 3.56+ε approximation by solving a strengthened configuration LP and applying a novel bipartite multigraph rounding. It introduces a global large-item concept, a greedy proxy function that is additive within configurations, and refined concentration bounds, with computer-assisted mathematical programs to bound the ratio. Beyond algorithmic gains, it establishes integrality-gap lower bounds for the configuration LP: at least $2^{\ln 2}-\varepsilon$ for weighted submodular valuations and $e^{1/e}-\varepsilon$ for additive valuations, plus a $2^{1/4}-\varepsilon$ gap for unweighted NSW with additive valuations. These results illuminate the limits of configuration-LP-based approaches and advance the understanding of NSW under submodular and additive valuations, with implications for fair and efficient resource allocation.

Abstract

We study the problem of allocating items to agents with submodular valuations with the goal of maximizing the weighted Nash social welfare (NSW). The best-known results for unweighted and weighted objectives are the $(4+ε)$ approximation given by Garg, Husic, Li, Végh, and Vondrák~[STOC 2023] and the $(233+ε)$ approximation given by Feng, Hu, Li, and Zhang~[STOC 2025], respectively. In this work, we present a $(3.56+ε)$-approximation algorithm for weighted NSW maximization with submodular valuations, simultaneously improving the previous approximation ratios of both the weighted and unweighted NSW problems. Our algorithm solves the configuration LP of Feng, Hu, Li, and Zhang~[STOC 2025] via a stronger separation oracle that loses an $e/(e-1)$ factor only on small items, and then rounds the solution via a new bipartite multigraph construction. Some key technical ingredients of our analysis include a greedy proxy function, additive within each configuration, that preserves the LP value while lower-bounding the rounded solution, together with refined concentration bounds and a series of mathematical programs analyzed partly by computer assistance. On the hardness side, we prove that the configuration LP for weighted NSW with submodular valuations has an integrality gap of at least $(2^{\ln 2}-ε) \approx 1.617 - ε$, which is larger than the current best-known $e/(e-1)-ε\approx 1.582-ε$ hardness~[SODA 2020]. For additive valuations, we show an integrality gap of $(e^{1/e}-ε)$, which proves the tightness of the approximation ratio in~[ICALP 2024] for algorithms based on the configuration LP. For unweighted NSW with additive valuations, we show an integrality gap of $(2^{1/4}-ε) \approx 1.189-ε$, again larger than the current best-known $\sqrt{8/7} \approx 1.069$-hardness~[Math. Oper. Res. 2024].

Figures (6)

• Figure 1: Illustration for the constructed bipartite multigraph. The figure only shows one agent $i$. We assume that one agent's valuation function is a coverage function, as shown in the subfigure (a), where each item is a set and the valuation of an item set is equal to the size of the union of these items. The subfigure (b) shows a fractional solution to \ref{['Conf-LP']}. Each rectangle represents a configuration, where the height is $v_i(S)$ and the width is $y^*_{i,S}$. Inside each configuration, items have been sorted in greedy order. The subfigure (c) shows how to build the bipartite multigraph. Each rectangle stands for a pair $(S,j)$, where the height is $\phi_{j}^{S}$ and the width is $y^*_{i,S}$; so every rectangle would correspond to one or two edges in the bipartite graph. The yellow and blue rectangles then correspond to marked and unmarked edges, respectively. The subfigure (d) shows the constructed bipartite multigraph, where solid and dashed edges are marked and unmarked edges, respectively.
• Figure 2: Illustration of the configuration type defined by the LP and rounding partitions, where "#" represents the number of items. It is possible that some sets of $S^{{\mathrm{em}}}$, $S^{{\mathrm{nem}}}$, $S^{\lg}$, $S^{{\mathrm{md}}}$, $S^{{\mathrm{sm}}}$ are empty, e.g., as shown in subfigure (b), $S^{{\mathrm{nem}}}$ and $S^{{\mathrm{sm}}}$ are empty.
• Figure 3: Overview of Analysis of (\ref{['mp:one']}). The Figures (a), (b), (c) are input distributions, and (d), (e) are output distributions. After a series of worst-case reductions, the input and output distributions finally become (c) and (e), respectively. In \ref{['subsec:computer-program']}, we compare (c) and (e) via a computer program.
• Figure 4: Illustration of the gap instance to \ref{['Conf-LP']} with $k = 5$ and $\lambda = 3$. Big and small squares denote the heavy and light agents, respectively, and big and small circles denote the large and small items, respectively.
• Figure 5: Illustration for the gap instance to \ref{['Conf-LP']} when the valuation function is additive. The large and small rectangles represent the heavy and light agents, respectively. The large and small circles represent the large and small items, which have values $t$ and $1$, respectively. For each heavy agent $i^{\mathrm{hv}}_p$, there is a group of private light agents $(i^{\mathrm{lt}}_{p,q})_{q\in[k]}$ and small items $M^{\mathrm{sm}}_p$. A line between an agent and an item indicates the item can be assigned to the agent (with a non-zero value).

Theorems & Definitions (59)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1: Monotone Submodular Function
• Definition 2.2: Greedy Order
• Definition 2.3: Multilinear Extension
• Definition 2.4: Concave Extension
• Lemma 2.5: vondrakthesis
• Lemma 2.6