Approximating the Nash Social Welfare with Budget-Additive Valuations

Garg, Jugal; Hoefer, Martin; Mehlhorn, Kurt

doi:10.1137/1.9781611975031.150

Cited by 51 publications

(45 citation statements)

References 76 publications

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“…This was improved to 2 by Cole et al [17], and further to 1.45 by Barman et al [10]. See also [4,5,23] for approximation algorithms in more general settings (with non-additive valuations). With the exception of [10] that uses item pricing techniques, rounding of convex programming relaxations is the main algorithmic tool in this line of research.…”

Section: Related Workmentioning

confidence: 98%

Envy-Freeness Up to Any Item with High Nash Welfare

Caragiannis

Gravin

Huang

2019

Proceedings of the 2019 ACM Conference on Economics and Computation

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Several fairness concepts have been proposed recently in attempts to approximate envy-freeness in settings with indivisible goods. Among them, the concept of envy-freeness up to any item (EFX) is arguably the closest to envy-freeness. Unfortunately, EFX allocations are not known to exist except in a few special cases. We make significant progress in this direction. We show that for every instance with additive valuations, there is an EFX allocation of a subset of items with a Nash welfare that is at least half of the maximum possible Nash welfare for the original set of items. That is, after donating some items to a charity, one can distribute the remaining items in a fair way with high efficiency. This bound is proved to be best possible. Our proof is constructive and highlights the importance of maximum Nash welfare allocation. Starting with such an allocation, our algorithm decides which items to donate and redistributes the initial bundles to the agents, eventually obtaining an allocation with the claimed efficiency guarantee. The application of our algorithm to large markets, where the valuations of an agent for every item is relatively small, yields EFX with almost optimal Nash welfare. To the best of our knowledge, this is the first use of large market assumptions in the fair division literature. We also show that our algorithm can be modified to compute, in polynomial-time, EFX allocations that approximate optimal Nash welfare within a factor of at most 2ρ, using a ρ-approximate allocation on input instead of the maximum Nash welfare one.

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Section: Related Workmentioning

confidence: 98%

Envy-Freeness Up to Any Item with High Nash Welfare

Caragiannis

Gravin

Huang

2019

Proceedings of the 2019 ACM Conference on Economics and Computation

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“…Algorithm Hardness Algorithm Restricted Additive 1.069 [GHM19] 1.45 [BKV18] [S] 1.069 [GHM19] O Table 1: Summary of results. Every entry has the best known approximation guarantee for the setting followed by the reference, from this paper or otherwise, that establishes it.…”

Section: Symmetric Agents Asymmetric Agents Hardnessmentioning

confidence: 99%

“…2 Observe that the partition problem reduces to the NSW problem with two identical agents. 3 Slight generalizations of additive valuations are studied: budget additive [GHM19], separable piecewise linear concave (SPLC) [AMGV18], and their combination [CCG + 18]. 4 For instance, the notion of a maximum bang-per-buck (MBB) item is critically used in most of these approaches.…”

Section: Introductionmentioning

confidence: 99%

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

Garg

Kulkarni

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study the problem of approximating maximum Nash social welfare (NSW) when allocating m indivisible items among n asymmetric agents with submodular valuations. The NSW is a well-established notion of fairness and efficiency, defined as the weighted geometric mean of agents' valuations. For special cases of the problem with symmetric agents and additive(-like) valuation functions, approximation algorithms have been designed using approaches customized for these specific settings, and they fail to extend to more general settings. Hence, no approximation algorithm with factor independent of m is known either for asymmetric agents with additive valuations or for symmetric agents beyond additive(-like) valuations.In this paper, we extend our understanding of the NSW problem to far more general settings. Our main contribution is two approximation algorithms for asymmetric agents with additive and submodular valuations respectively. Both algorithms are simple to understand and involve nontrivial modifications of a greedy repeated matchings approach. Allocations of high valued items are done separately by un-matching certain items and re-matching them, by processes that are different in both algorithms. We show that these approaches achieve approximation factors of O(n) and O(n log n) for additive and submodular case respectively, which is independent of the number of items. For additive valuations, our algorithm outputs an allocation that also achieves the fairness property of envy-free up to one item (EF1).Furthermore, we show that the NSW problem under submodular valuations is strictly harder than all currently known settings with an e e−1 factor of the hardness of approximation, even for constantly many agents. For this case, we provide a different approximation algorithm that achieves a factor of e e−1 , hence resolving it completely.

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“…Therefore, the Nash welfare of an allocation is also considered as a measure of efficiency and fairness of an allocation. However, finding an allocation with the maximum Nash welfare is APX-hard [25], and its approximation has received a lot of attention recently, e.g., [2,3,5,11,[14][15][16][17]. Barman et al [5] give a pseudopolynomial algorithm to find an allocation that is both EF1 and Pareto-optimal.…”

Section: Further Related Workmentioning

confidence: 99%

EFX Exists for Three Agents

Chaudhury

Garg

Mehlhorn

2020

Proceedings of the 21st ACM Conference on Economics and Computation

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We study the problem of distributing a set of indivisible items among agents with additive valuations in a fair manner. The fairness notion under consideration is Envy-freeness up to any item (EFX). Despite significant efforts by many researchers for several years, the existence of EFX allocations has not been settled beyond the simple case of two agents. In this paper, we show constructively that an EFX allocation always exists for three agents. Furthermore, we falsify the conjecture by Caragiannis et al. [9] by showing an instance with three agents for which there is a partial EFX allocation (some items are not allocated) with higher Nash welfare than that of any complete EFX allocation. CCS Concepts: • Theory of computation → Algorithmic game theory.

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Approximating the Nash Social Welfare with Budget-Additive Valuations

Cited by 51 publications

References 76 publications

Envy-Freeness Up to Any Item with High Nash Welfare

Envy-Freeness Up to Any Item with High Nash Welfare

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

EFX Exists for Three Agents

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