Matching problems with group-fairness constraints and diversity constraints have numerous applications such as in allocation problems, committee selection, school choice, etc. Moreover, online matching problems have lots of applications in ad allocations and other e-commerce problems like product recommendation in digital marketing. We study two problems involving assigning {\em items} to {\em platforms}, where items belong to various {\em groups} depending on their attributes; the set of items are available offline and the platforms arrive online. In the first problem, we study online matchings with {\em proportional fairness constraints}. Here, each platform on arrival should either be assigned a set of items in which the fraction of items from each group is within specified bounds or be assigned no items; the goal is to assign items to platforms in order to maximize the number of items assigned to platforms. In the second problem, we study online matchings with {\em diversity constraints}, i.e. for each platform, absolute lower bounds are specified for each group. Each platform on arrival should either be assigned a set of items that satisfy these bounds or be assigned no items; the goal is to maximize the set of platforms that get matched. We study approximation algorithms and hardness results for these problems. The technical core of our proofs is a new connection between these problems and the problem of matchings in hypergraphs. Our experimental evaluation shows the performance of our algorithms on real-world and synthetic datasets exceeds our theoretical guarantees.

翻译：具有群体公平约束和多样性约束的匹配问题在分配问题、委员会选举、学校选择等领域具有广泛应用。此外，在线匹配问题在广告分配及其他电子商务问题（如数字营销中的产品推荐）中也有大量应用。我们研究了将**物品**分配给**平台**的两个问题，其中物品根据其属性属于不同**群体**；物品集合离线可用，而平台在线到达。在第一个问题中，我们研究具有**比例公平约束**的在线匹配。此处，每个到达的平台要么被分配一组物品，其中来自每个群体的物品比例在指定范围内，要么不分配任何物品；目标是通过向平台分配物品来最大化分配物品的总数。在第二个问题中，我们研究具有**多样性约束**的在线匹配，即对每个平台指定每个群体的绝对下限。每个到达的平台要么被分配一组满足这些约束的物品，要么不分配任何物品；目标是最大化成功匹配的平台集合。我们研究了这些问题的近似算法和硬度结果。我们证明的技术核心是这些问题与超图中匹配问题之间的新联系。我们的实验评估表明，算法在真实和合成数据集上的性能超越了理论保证。