Fair division is a fundamental problem in various multi-agent settings, where the goal is to divide a set of resources among agents in a fair manner. We study the case where m indivisible items need to be divided among n agents with additive valuations using the popular fairness notion of maximin share (MMS). An MMS allocation provides each agent a bundle worth at least her maximin share. While it is known that such an allocation need not exist [1], [2], a series of remarkable work [1], [3], [4], [5], [6] provided approximation algorithms for a $\frac{2}{3}$-MMS allocation in which each agent receives a bundle worth at least $\frac{2}{3}$ times her maximin share. More recently, Ghodsi et al. [7] showed the existence of a $\frac{3}{4}$-MMS allocation and a PTAS to find a ($\frac{3}{4}-ϵ$)-MMS allocation for an $ϵ>0$. Most of the previous works utilize intricate algorithms and require agents' approximate MMS values, which are computationally expensive to obtain.

In this paper, we develop a new approach that gives a simple algorithm for showing the existence of a $\frac{3}{4}$-MMS allocation. Furthermore, our approach is powerful enough to be easily extended in two directions: First, we get a strongly polynomial time algorithm to find a $\frac{3}{4}$-MMS allocation, where we do not need to approximate the MMS values at all. Second, we show that there always exists a $(\frac{3}{4}+\frac{1}{12n})$-MMS allocation, improving the best previous factor. This improves the approximation guarantee, most notably for small n. We note that $\frac{3}{4}$ was the best factor known for $n>4$.

公平划分是各种多代理设置中的基本问题，其目标是以公平的方式在代理之间分配一组资源。我们研究了m 个不可分割的项目需要在n 个代理之间使用最大共享(MMS)的流行公平概念进行附加估值的情况。MMS 分配为每个代理提供至少价值其最大份额的捆绑包。虽然已知这种分配不一定存在 [1]、[2]，但一系列出色的工作 [1]、[3]、[4]、[5]、[6] 提供了近似算法$\frac{2}{3}$- MMS 分配，其中每个代理收到至少价值的捆绑包 $\frac{2}{3}$乘以她的最大份额。最近，Ghodsi 等人。[7] 表明存在一个$\frac{3}{4}$-MMS 分配和 PTAS 以查找 ($\frac{3}{4}-\epsilon$)-MMS 分配 $\epsilon >0$. 以前的大部分工作都使用复杂的算法，并且需要代理的近似 MMS 值，获得这些值的计算成本很高。

在本文中，我们开发了一种新方法，该方法提供了一种简单的算法来显示存在 $\frac{3}{4}$-彩信分配。此外，我们的方法足够强大，可以很容易地在两个方向上扩展：首先，我们得到一个强多项式时间算法来找到一个$\frac{3}{4}$-MMS 分配，我们根本不需要近似 MMS 值。其次，我们证明总是存在一个$(\frac{3}{4}+\frac{1}{12n})$- 彩信分配，提高最佳前因数。这提高了近似保证，尤其是对于小n。我们注意到$\frac{3}{4}$ 是众所周知的最好的因素 $n>4$.