We study the problem of fair allocation of a set of indivisible items among agents with additive valuations, under cardinality constraints. In this setting the items are partitioned into categories, each with its own limit on the number of items it may contribute to any bundle. One example of such a problem is allocating seats in a multitrack conference. We consider the fairness measure known as the maximin share (MMS) guarantee, and propose a novel polynomial-time algorithm for finding $1/2$-approximate MMS allocations. We extend the notions and algorithms related to ordered and reduced instances to work with cardinality constraints, and combine these with a bag filling style procedure. Our algorithm improves on that of Biswas and Barman (IJCAI-18), with its approximation ratio of $1/3$. We also present an optimizing algorithm, which for each instance, instead of fixing $\alpha = 1/2$, uses bisection to find the largest $\alpha$ for which our algorithm obtains a valid $\alpha$-approximate MMS allocation. Numerical tests show that our algorithm finds strictly better approximations than the guarantee of $1/2$ for most instances, in many cases surpassing $3/5$. The optimizing version of the algorithm produces MMS allocations in a comparable number of instances to that of Biswas and Barman's algorithm, on average achieving a better approximation when MMS is not obtained.

中文翻译：

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在基数约束下保证半最大值份额

我们研究了在基数约束下，在具有附加估值的代理之间公平分配一组不可分割项目的问题。在此设置中，项目被划分为多个类别，每个类别对它可能贡献给任何捆绑的项目数量都有自己的限制。此类问题的一个示例是在多轨会议中分配席位。我们考虑了称为最大最小份额 (MMS) 保证的公平性度量，并提出了一种新的多项式时间算法来寻找 $1/2$ 近似的 MMS 分配。我们扩展了与有序实例和缩减实例相关的概念和算法以处理基数约束，并将它们与袋子填充样式程序相结合。我们的算法改进了 Biswas 和 Barman (IJCAI-18) 的算法，其近似比率为 $1/3$。我们还提出了一个优化算法，对于每个实例，它不是固定 $\alpha = 1/2$，而是使用二分法来找到最大的 $\alpha$，我们的算法为其获得有效的 $\alpha$-approximate MMS 分配。数值测试表明，我们的算法在大多数情况下找到了比 $1/2$ 保证更好的近似值，在许多情况下超过 $3/5$。该算法的优化版本在与 Biswas 和 Barman 算法相当数量的实例中产生 MMS 分配，平均而言，在未获得 MMS 时实现更好的近似。在许多情况下超过 $3/5$。该算法的优化版本在与 Biswas 和 Barman 算法相当数量的实例中产生 MMS 分配，平均而言，在未获得 MMS 时实现更好的近似。在许多情况下超过 $3/5$。该算法的优化版本在与 Biswas 和 Barman 算法相当数量的实例中产生 MMS 分配，平均而言，在未获得 MMS 时实现更好的近似。