Simplification and Improvement of MMS Approximation

Abstract

We consider the problem of fairly allocating a set of indivisible goods among n agents with additive valuations, using the popular fairness notion of maximin share (MMS). Since MMS allocations do not always exist, a series of works provided existence and algorithms for approximate MMS allocations. The Garg-Taki algorithm gives the current best approximation factor of (3/4 + 1/12n). Most of these results are based on complicated analyses, especially those providing better than 2/3 factor. Moreover, since no tight example is known of the Garg-Taki algorithm, it is unclear if this is the best factor of this approach. In this paper, we significantly simplify the analysis of this algorithm and also improve the existence guarantee to a factor of (3/4 + min(1/36, 3/(16n-4))). For small n, this provides a noticeable improvement. Furthermore, we present a tight example of this algorithm, showing that this may be the best factor one can hope for with the current techniques.

The budgeted maximin share allocation problem

“…If b i = +∞ for each agent i, the budgeted MMS allocation problem is exactly the MMS allocation [12]. Kurokawa et al [26] demonstrated that the exact MMS may not exist, but 2 3 -approximate MMS can guarantee its existence. Amanatidis et al [5] proposed a ( 23 − ǫ)-approximation guaranteed polynomial time algorithm.…”

“…Garg et al [18] proposed a ( ). Furthermore, Akrami et al [2] proved the existence of a randomized allocation such that each agent obtains 3 4 her MMS value (ex-post) and ( 17√ 3 − 24)/4 √ 3 her MMS value (ex-ante). If s(g) = 1 for each good g, the budgeted MMS allocation problem is exactly the MMS allocation under cardinality constraints.…”

“…Hummel et. al [24] improved the existence guarantee to 2 3 and proposed a 2 3 -approximation guaranteed polynomial time algorithm.…”

1/2-Approximate MMS Allocation for Separable Piecewise Linear Concave Valuations

“…We consider the problem of fairly allocating a set M of m indivisible goods among a set N of n agents with heterogeneous preferences under the popular fairness notions of Maximin share (MMS) (Budish 2011) and Any Price Share (APS) (Babaioff, Ezra, and Feige 2021). These notions have been extensively studied for the setting where the agents have additive valuations (Barman and Krishna Murthy 2017;Ghodsi et al 2018;Garg, McGlaughlin, and Taki 2018;Garg and Taki 2020;Akrami, Garg, and Taki 2023;. This paper studies the problem beyond additive valuations, particularly for the classical separable-concave valuations (Vazirani and Yannakakis 2011;Chaudhury et al 2022) and submodular valuations (Ghodsi et al 2018; Barman and Krishnamurthy 2020; Uziahu and Feige 2023).…”