Abstract: In this work, we study the maximin share fairness notion for allocation of indivisible goods in the subadditive and fractionally subadditive settings. While previous work refutes the possibility of obtaining an allocation which is better than $1/2$-\textsf{MMS}, the only positive result for the subadditive setting states that when the number of items is equal to $m$, there always exists an $\Omega(1/\log m)$-\textsf{MMS} allocation. Since the number of items may be larger than the number of agents ($n$), such a bound can only imply a weak bound of $\Omega(\frac{1}{n \log n})$-\textsf{MMS} allocation in general.



In this work, we improve this gap exponentially to an $\Omega(\frac{1}{\log n \log \log n})$-\textsf{MMS} guarantee. In addition to this, we prove that when the valuation functions are fractionally subadditive, a $1/4.6$-\textsf{MMS} allocation is guaranteed to exist. This also improves upon the previous bound of $1/5$-\textsf{MMS} guarantee for the fractionally subadditive setting.