Approximation Algorithms for Maximin Fair Division

Barman, S and Krishnamurthy, SK (2020) Approximation Algorithms for Maximin Fair Division. In: ACM Transactions on Economics and Computation, 8 (1).

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Abstract

We consider the problem of allocating indivisible goods fairly among n agents who have additive and submodular valuations for the goods. Our fairness guarantees are in terms of the maximin share, which is defined to be the maximum value that an agent can ensure for herself, if she were to partition the goods into n bundles, and then receive a minimum valued bundle. Since maximin fair allocations (i.e., allocations in which each agent gets at least her maximin share) do not always exist, prior work has focused on approximation results that aim to find allocations in which the value of the bundle allocated to each agent is (multiplicatively) as close to her maximin share as possible. In particular, Procaccia and Wang (2014) along with Amanatidis et al. (2015) have shown that under additive valuations, a 2/3-Approximate maximin fair allocation always exists and can be found in polynomial time. We complement these results by developing a simple and efficient algorithm that achieves the same approximation guarantee. Furthermore, we initiate the study of approximate maximin fair division under submodular valuations. Specifically, we show that when the valuations of the agents are nonnegative, monotone, and submodular, then a 0.21-Approximate maximin fair allocation is guaranteed to exist. In fact, we show that such an allocation can be efficiently found by using a simple round-robin algorithm. A technical contribution of the article is to analyze the performance of this combinatorial algorithm by employing the concept of multilinear extensions. © 2020 ACM.

Item Type:	Journal Article
Publication:	ACM Transactions on Economics and Computation
Publisher:	Association for Computing Machinery
Additional Information:	Copy right for this article belongs to Association for Computing Machinery
Keywords:	Additives; Polynomial approximation; Scheduling algorithms, Approximation results; Combinatorial algorithm; Fairness guarantee; Indivisible good; Multilinear extension; Round Robin algorithms; Simple and efficient algorithms; Technical contribution, Approximation algorithms
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	05 Nov 2020 09:08
Last Modified:	05 Nov 2020 09:08
URI:	http://eprints.iisc.ac.in/id/eprint/65811

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