# Tight Approximation Algorithms for p-Mean Welfare Under Subadditive Valuations

arXiv: https://arxiv.org/abs/2005.07370 · HTML: https://mlsanity.com/paper/2005.07370/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 May 2020.

## Built on by (most cited first, 8 of 8)

- [Temporal Fair Division of Indivisible Items](https://mlsanity.com/paper/2410.14593/) (2024)
- [p-Mean Regret for Stochastic Bandits](https://mlsanity.com/paper/2412.10751/) (2024)
- [Revisiting Social Welfare in Bandits: UCB is (Nearly) All You Need](https://mlsanity.com/paper/2510.21312/) (2025)
- [Improved Algorithms for Nash Welfare in Linear Bandits](https://mlsanity.com/paper/2601.22969/) (2026)
- [The Cost of EFX: Generalized-Mean Welfare and Complexity Dichotomies with Few Surplus Items](https://mlsanity.com/paper/2601.12849/) (2026)
- [Price of Fairness in Bandits: A Tight Minimax Characterization](https://mlsanity.com/paper/2607.13402/) (2026)
- [Dividing Good and Better Items Among Agents with Bivalued Submodular Valuations](https://mlsanity.com/paper/2302.03087/) (2023)
- [To Pool or Not To Pool: Analyzing the Regularizing Effects of Group-Fair Training on Shared Models](https://mlsanity.com/paper/2402.18803/) (2024)

## Abstract (arXiv)

We develop polynomial-time algorithms for the fair and efficient allocation of indivisible goods among n agents that have subadditive valuations over the goods. We first consider the Nash social welfare as our objective and design a polynomial-time algorithm that, in the value oracle model, finds an 8n-approximation to the Nash optimal allocation. Subadditive valuations include XOS (fractionally subadditive) and submodular valuations as special cases. Our result, even for the special case of submodular valuations, improves upon the previously best known O(n log n)-approximation ratio of Garg et al. (2020). More generally, we study maximization of p-mean welfare. The p-mean welfare is parameterized by an exponent term p ∈ (-∞, 1] and encompasses a range of welfare functions, such as social welfare (p = 1), Nash social welfare (p → 0), and egalitarian welfare (p → -∞). We give an algorithm that, for subadditive valuations and any given p ∈ (-∞, 1], computes (in the value oracle model and in polynomial time) an allocation with p-mean welfare at least frac18n times the optimal. Further, we show that our approximation guarantees are essentially tight for XOS and, hence, subadditive valuations. We adapt a result of Dobzinski et al. (2010) to show that, under XOS valuations, an O left(n1-varepsilon right) approximation for the p-mean welfare for any p ∈ (-∞,1] (including the Nash social welfare) requires exponentially many value queries; here, varepsilon>0 is any fixed constant.