  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, a series of
work provided approximation algorithms for a 2/3-MMS allocation in which each
agent receives a bundle worth at least 2/3 times her maximin share. More
recently, Ghodsi et al. [EC'2018] showed the existence of a 3/4-MMS allocation
and a PTAS to find a (3/4-\epsilon)-MMS allocation for an \epsilon > 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 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 3/4-MMS allocation, where we do
not need to approximate the MMS values at all. Second, we show that there
always exists a (3/4 + 1/(12n))-MMS allocation, improving the best previous
factor. This improves the approximation guarantee, most notably for small n. We
note that 3/4 was the best factor known for n> 4.
