An improved approximation algorithm for maximin shares

Fair division is a fundamental problem in various multi-agent settings, where the goal is to divide a set of resources among agents in a fair manner. We study the case where m indivisible items need to be divided among n agents with additive valuations using the popular fairness notion of maximin share (MMS). An MMS allocation provides each agent a bundle worth at least her maximin share. While it is known that such an allocation need not exist [1,2], a series of remarkable work [1,3-6] provided approximation algorithms for a 2/3-MMS allocation in which each agent receives a bundle worth at least 2/3 times her maximin share. More recently, Ghodsi et al. [7] showed the existence of a 3/4-MMS allocation and a PTAS to find a (3/4 - epsilon )-MMS allocation for an epsilon > 0. Most of the previous works utilize intricate algorithms and require agents' approximate MMS values, which are computationally expensive to obtain. In this paper, we develop a new approach that gives a simple algorithm for showing the existence of a 3/4-MMS allocation. Furthermore, our approach is powerful enough to be easily extended in two directions: First, we get a strongly polynomial time algorithm to find a 3/4 -MMS allocation, where we do not need to approximate the MMS values at all. Second, we show that there always exists a (3/4 + 1/12n)-MMS allocation, improving the best previous factor. This improves the approximation guarantee, most notably for small n. We note that 3/4 was the best factor known for n > 4. (C) 2021 Elsevier B.V. All rights reserved.