Maximin Shares with Lower Quotas

We study the fair division of indivisible items among $n$ agents with heterogeneous additive valuations, subject to lower and upper quotas on the number of items allocated to each agent. Such constraints are crucial in various applications, ranging from personnel assignments to computing resource distribution. This paper focuses on the fairness criterion known as maximin shares (MMS) and its approximations. Under arbitrary lower and upper quotas, we show that a $\left(\frac{2n}{3n-1}\right)$-MMS allocation of goods exists and can be computed in polynomial time, while we also present a polynomial-time algorithm for finding a $\left(\frac{3n-1}{2n}\right)$-MMS allocation of chores. Furthermore, we consider the generalized scenario where items are partitioned into multiple categories, each with its own lower and upper quotas. In this setting, our algorithm computes an $\left(\frac{n}{2n-1}\right)$-MMS allocation of goods or a $\left(\frac{2n-1}{n}\right)$-MMS allocation of chores in polynomial time. These results extend previous work on the cardinality constraints, i.e., the special case where only upper quotas are imposed.