# Approximating the Nash Social Welfare with Budget-Additive Valuations

@article{Garg2018ApproximatingTN, title={Approximating the Nash Social Welfare with Budget-Additive Valuations}, author={Jugal Garg and Martin Hoefer and Kurt Mehlhorn}, journal={ArXiv}, year={2018}, volume={abs/1707.04428} }

We present the first constant-factor approximation algorithm for maximizing the Nash social welfare when allocating indivisible items to agents with budget-additive valuation functions. Budget-additive valuations represent an important class of submodular functions. They have attracted a lot of research interest in recent years due to many interesting applications. For every $\varepsilon > 0$, our algorithm obtains a $(2.404 + \varepsilon)$-approximation in time polynomial in the input size and…

## 48 Citations

Sublinear Approximation Algorithm for Nash Social Welfare with XOS Valuations

- Economics, Computer ScienceArXiv
- 2021

This work breaks the O(n)approximation barrier for NSW maximization under XOS valuations by developing a novel connection between NSW and social welfare under a capped version of the agents’ valuations.

Satiation in Fisher Markets and Approximation of Nash Social Welfare.

- Economics
- 2017

We study linear Fisher markets with satiation. In these markets, sellers have earning limits and buyers have utility limits. Beyond natural applications in economics, these markets arise in the…

Fair Division of Indivisible Goods for a Class of Concave Valuations

- Economics, Computer ScienceJournal of Artificial Intelligence Research
- 2022

A polynomial-time algorithm is presented that approximates the optimal Nash social welfare (NSW) up to a factor of e1/e ≈ 1.445, and the upper bounds on the optimal NSW introduced in Cole and Gkatzelis (2018) and Barman et al. ( 2018) have the same value.

Approximating Nash social welfare under rado valuations

- Computer ScienceSTOC
- 2021

The approach gives the first constant-factor approximation algorithm for the asymmetric case under Rado valuations, provided that the maximum ratio between the weights is bounded by a constant.

Approximating Nash Social Welfare under Submodular Valuations through (Un)Matchings

- Computer ScienceSODA
- 2020

The problem of approximating maximum Nash social welfare when allocating m indivisible items among n asymmetric agents with submodular valuations is extended to far more general settings and is shown to be strictly harder than all currently known settings with an e/(e-1) factor of the hardness of approximation.

Online Nash Social Welfare via Promised Utilities.

- Computer Science, Economics
- 2020

This work provides an online algorithm that achieves a competitive ratio of $O(\log N) and $O(log T)$, but also a stronger competitive ratios in settings where the value of any agent for her most preferred item is no more than $k$ times her average value.

An Additive Approximation Scheme for the Nash Social Welfare Maximization with Identical Additive Valuations

- EconomicsArXiv
- 2022

A polynomial-time algorithm is given that maximizes the Nash social welfare within an additive error εvmax, where ε is an arbitrary positive number and vmax is the maximum utility of a good.

Greedy Algorithms for Maximizing Nash Social Welfare

- EconomicsAAMAS
- 2018

The effectiveness of simple, greedy algorithms in solving the problem of fairly allocating a set of indivisible goods among agents with additive valuations is studied, showing that when agents have binary valuations over the goods, an exact solution can be found in polynomial time via a greedy algorithm.

Estimating the Nash Social Welfare for coverage and other submodular valuations

- Economics, MathematicsSODA
- 2021

This work provides a 1 e (1 − 1 e )-approximation of the optimal value for several classes of submodular valuations: coverage, sums of matroid rank functions, and certain matching-based valuations.

Approximating Nash Social Welfare in 2-Valued Instances

- Mathematics, EconomicsArXiv
- 2021

It is shown that an optimal allocation can be computed in polynomial time if p divides q, and an APX-hardness result is proved for the problem with a lower bound on the ratio of 1.033.

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