  We consider the problem of allocating indivisible goods fairly among n agents
who have additive and submodular valuations for the goods. Our fairness
guarantees are in terms of the maximin share, that is defined to be the maximum
value that an agent can ensure for herself, if she were to partition the goods
into n bundles, and then receive a minimum valued bundle. Since maximin fair
allocations (i.e., allocations in which each agent gets at least her maximin
share) do not always exist, prior work has focused on approximation results
that aim to find allocations in which the value of the bundle allocated to each
agent is (multiplicatively) as close to her maximin share as possible. In
particular, Procaccia and Wang (2014) along with Amanatidis et al. (2015) have
shown that under additive valuations a 2/3-approximate maximin fair allocation
always exists and can be found in polynomial time. We complement these results
by developing a simple and efficient algorithm that achieves the same
approximation guarantee.
  Furthermore, we initiate the study of approximate maximin fair division under
submodular valuations. Specifically, we show that when the valuations of the
agents are nonnegative, monotone, and submodular, then a 0.21-approximate
maximin fair allocation is guaranteed to exist. In fact, we show that such an
allocation can be efficiently found by using a simple round-robin algorithm. A
technical contribution of the paper is to analyze the performance of this
combinatorial algorithm by employing the concept of multilinear extensions.
