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 [Formula presented]-MMS allocation in which each agent receives a bundle worth at least [Formula presented] times her maximin share. More recently, Ghodsi et al. [7] showed the existence of a [Formula presented]-MMS allocation and a PTAS to find a ([Formula presented])-MMS allocation for an ϵ>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 [Formula presented]-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 [Formula presented]-MMS allocation, where we do not need to approximate the MMS values at all. Second, we show that there always exists a [Formula presented]-MMS allocation, improving the best previous factor. This improves the approximation guarantee, most notably for small n. We note that [Formula presented] was the best factor known for n>4.