  We consider the problem of fairly allocating a set of indivisible items under
the criteria of the maximin share guarantee. Specifically, we study
approximation of maximin share allocations under hereditary set system
valuations, in which each valuation function is based on the independent sets
of an underlying hereditary set systems. Using a lone divider approach, we show
the existence of $1/2$-approximate MMS allocations, improving on the $11/30$
guarantee of Li and Vetta. Moreover, we prove that ($2/3 +
\epsilon$)-approximate MMS allocations do not always exist in this model for
every $\epsilon > 0$, an improvement from the recent $3/4 + \epsilon$ result of
Li and Deng. Our existence proof is constructive, but does not directly yield a
polynomial-time approximation algorithm. However, we show that a
$2/5$-approximate MMS allocation can be found in polynomial time, given
valuation oracles. Finally, we show that our existence and approximation
results transfer to a variety of problems within constrained fair allocation,
improving on existing results in some of these settings.
