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 current best approximation factor, for which the existence is known, is $(\frac{3}{4} + \frac{1}{12n})$ [Garg and Taki, 2021]. 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 $(\frac{3}{4} + \min(\frac{1}{36}, \frac{3}{16n-4}))$. 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.

翻译：本文研究了在加性估值下，将一组不可分割物品公平分配给$n$个智能体的问题，采用流行的公平性概念——最大最小份额（MMS）。由于MMS分配并非始终存在，一系列研究提出了近似MMS分配的存在性及相应算法。目前已知存在性的最佳近似因子为$(\frac{3}{4} + \frac{1}{12n})$ [Garg and Taki, 2021]。这些结果大多基于复杂的分析，尤其是那些因子优于$2/3$的结果。此外，由于Garg-Taki算法尚无紧实例，该方法是否能达到该最佳因子尚不明确。本文显著简化了该算法的分析过程，并将存在性保证改进至因子$(\frac{3}{4} + \min(\frac{1}{36}, \frac{3}{16n-4}))$。对于较小的$n$，这一改进尤为显著。此外，我们给出了该算法的紧实例，表明在当前技术条件下，这可能是可以达到的最佳因子。