In this work, we study the maximin share fairness notion ($\mathsf{\text{MMS}}$) 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-MMS, the only positive result for the subadditive setting states that when the number of items is equal to m, there always exists an $\mathrm{\Omega }(1/\log m)$-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 $\mathrm{\Omega }(\frac{1}{n\log n})$-$\mathsf{\text{MMS}}$ allocation in general.

In this work, we improve this bound exponentially to $\mathrm{\Omega }(\frac{1}{\log n\log \log n})$-MMS guarantee. In addition to this, we prove that when the valuation functions are fractionally subadditive, a 0.2192235-MMS allocation is guaranteed to exist. This also improves upon the previous bound of 1/5-MMS guarantee for the fractionally subadditive setting.

在这项工作中，我们研究了最大最小份额公平概念（$\mathsf{\text{彩信}}$) 用于在次加法和部分次加法设置中分配不可分割的商品。虽然以前的工作驳斥了获得优于 1/2-MMS 的分配的可能性，但次加性设置的唯一积极结果表明，当项目数等于 m，总是存在 $\mathrm{\Omega }（1/\mathrm{日志}米）$-彩信分配。由于项目的数量可能大于代理的数量（n），因此这样的界限只能暗示的弱界限$\mathrm{\Omega }（\frac{1}{n\mathrm{日志}n}）$-$\mathsf{\text{彩信}}$分配一般。

在这项工作中，我们将这个界限以指数方式改进为$\mathrm{\Omega }（\frac{1}{\mathrm{日志}n\mathrm{日志}\mathrm{日志}n}）$-彩信保证。除此之外，我们还证明，当估值函数部分可加时，保证存在 0.2192235-MMS 分配。这也改进了之前分数次加性设置的 1/5-MMS 保证界限。