We study the fundamental problem of fairly allocating a set of indivisible goods among $n$ agents with additive valuations using the desirable fairness notion of maximin share (MMS). MMS is the most popular share-based notion, in which an agent finds an allocation fair to her if she receives goods worth at least her MMS value. An allocation is called MMS if all agents receive at least their MMS value. Since MMS allocations need not exist when $n>2$, a series of works showed the existence of approximate MMS allocations with the current best factor of $\frac34 + O(\frac{1}{n})$. However, a simple example in [DFL82, BEF21, AGST23] showed the limitations of existing approaches and proved that they cannot improve this factor to $3/4 + \Omega(1)$. In this paper, we bypass these barriers to show the existence of $(\frac{3}{4} + \frac{3}{3836})$-MMS allocations by developing new reduction rules and analysis techniques.

翻译：我们研究在加性估值下将一组不可分割物品公平分配给$n$个代理人的基本问题，采用最大最小份额（MMS）这一理想公平性概念。MMS是最流行的份额概念，当代理人获得至少等于其MMS价值的物品时，她认为分配是公平的。如果所有代理人都获得至少其MMS价值，则分配称为MMS。由于当$n>2$时MMS分配不一定存在，一系列工作证明了近似MMS分配的存在性，当前最佳因子为$\frac34 + O(\frac{1}{n})$。然而，[DFL82, BEF21, AGST23]中的一个简单例子显示了现有方法的局限性，并证明它们无法将此因子改进到$3/4 + \Omega(1)$。在本文中，我们通过开发新的归约规则和分析技术，绕过这些障碍，证明了$(\frac{3}{4} + \frac{3}{3836})$-MMS分配的存在性。