Maximin Shares in Hereditary Set Systems

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.