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. However, 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 $\frac{3}{4} + O(\frac{1}{n})$. The recent work by Akrami et al. 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：我们研究不可分割物品公平分配的基本问题，考虑$n$个具有可加估值的智能体，并采用理想公平概念——最大最小份额（MMS）。MMS是最流行的基于份额的概念，在该概念中，若智能体获得的物品价值至少等于其MMS值，则该分配对其公平。当所有智能体获得至少其MMS值，该分配称为MMS分配。然而，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$\frac{3}{4} + O(\frac{1}{n})$。Akrami等人近期工作揭示了现有方法的局限性，并证明这些方法无法将该因子改进至$3/4 + \Omega(1)$。在本文中，我们通过开发新的缩减规则和分析技术，绕开这些障碍，证明了$(\frac{3}{4} + \frac{3}{3836})$-MMS分配的存在性。