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 $(\frac{3}{4} + \frac{1}{12n})$. 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 $(\frac{3}{4} + \min(\frac{1}{36}, \frac{3}{16n-4}))$. 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：我们考虑在加性估值函数下，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——最大化最小份额（MMS）。由于MMS分配并非总是存在，一系列研究提供了近似MMS分配的存在性证明与算法。Garg-Taki算法给出了当前最佳近似因子$(\frac{3}{4} + \frac{1}{12n})$。这些结果大多基于复杂分析，尤其是那些因子优于$2/3$的成果。此外，由于Garg-Taki算法尚无紧例，该方法的最优因子尚不明确。本文显著简化了该算法的分析，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$n$，这提供了显著提升。此外，我们给出了该算法的一个紧例，表明这可能是在当前技术下能期待的最佳因子。