We initiate the study of fairness among of agents in online bipartite matching where there is a given set of offline vertices (aka agents) and another set of vertices (aka items) that arrive online and must be matched irrevocably upon arrival. In this setting, agents are partitioned into classes and the matching is required to be fair with respect to the classes. We adapt popular fairness notions (e.g. envy-freeness, proportionality, and maximin share) and their relaxations to this setting and study deterministic algorithms for matching indivisible items (leading to integral matchings) and for matching divisible items (leading to fractional matchings). For matching indivisible items, we propose an adaptive-priority-based algorithm, , prove that it achieves -approximation of both class envy-freeness up to one item and class maximin share fairness, and show that each guarantee is tight. For matching divisible items, we design a water-filling-based algorithm, , that achieves -approximation of class envy-freeness and class proportionality; we prove to be tight for class proportionality and establish a upper bound on class envy-freeness. Finally, we discuss several challenges in designing randomized algorithms that achieve reasonable fairness approximation ratios. Nonetheless, we build upon to design a randomized algorithm for matching indivisible items, , which achieves 0.593-approximation of class proportionality.

中文翻译：

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在线匹配的班级公平性


我们发起了在线二分匹配中代理之间公平性的研究，其中存在一组给定的离线顶点（又名代理）和另一组到达在线的顶点（又名项目），并且在到达时必须不可撤销地匹配。在此设置中，代理被划分为类别，并且要求匹配对于类别而言是公平的。我们采用流行的公平概念（例如无嫉妒、比例和最大最小份额）及其松弛来适应这种设置，并研究用于匹配不可分割项目（导致整数匹配）和匹配可分割项目（导致分数匹配）的确定性算法。对于匹配不可分割的项目，我们提出了一种基于自适应优先级的算法，证明它实现了一个项目的类嫉妒自由度和类最大最小共享公平性的近似，并表明每个保证都是严格的。为了匹配可整除的项目，我们设计了一种基于注水的算法 ，它实现了类嫉妒自由度和类比例性的近似；我们证明了阶级比例是严格的，并建立了阶级无嫉妒的上限。最后，我们讨论了设计实现合理公平近似比的随机算法的几个挑战。尽管如此，我们还是设计了一种随机算法来匹配不可分割的项目 ，它实现了类比例的 0.593 近似值。