In typical online matching problems, the goal is to maximize the number of matches. This paper studies online bipartite matching from the perspective of group-level fairness, and the goal is to balance the number of matches made across different groups of online nodes. We assume that an online node's group belongings are revealed upon arrival, and measure performance based on the group with the smallest fraction of its nodes matched at the end. We distinguish between two different objectives: long-run fairness, where the algorithm is audited for its fairness over many realizations from the same instance; and short-run fairness, where the algorithm is audited for its fairness on a single realization. We focus on the known-IID model of online arrivals and analyze, under both objectives, the competitive ratio for two classes of algorithms: non-rejecting algorithms, which must match an online node as long as a neighbor is available; and general online algorithms, which are allowed to reject online nodes to improve fairness. For long-run fairness, we analyze two online algorithms (sampling and pooling) which establish asymptotic optimality across many different regimes (no specialized supplies, no rare demand types, or imbalanced supply/demand). By contrast, outside all of these regimes, we show that the competitive ratio for online algorithms is between 0.632 and 0.732. For short-run fairness, we focus on the case of a complete bipartite graph and show that the competitive ratio for online algorithms is between 0.863 and 0.942; we also derive a probabilistic rejection algorithm which is asymptotically optimal as total demand increases.
翻译:在典型的在线匹配问题中,目标是最大限度地增加匹配数量。 本文从集团公平的角度对在线双偏比进行在线双向匹配, 目标是平衡不同组的在线节点匹配数量。 我们假设一个在线节点的群落属性在抵达时会披露, 并测量基于该组的性能, 其节点中最小的部分在结束时会匹配。 我们区分了两个不同的目标: 长期公平, 算法因其公平性而被审计, 相对于同一实例的许多实现; 短期公平性, 以及短期公平性, 算法被审计, 算法在单一实现中是否公平。 我们侧重于已知的在线抵达的二维正比模型, 并在这两个目标下分析两类算法的竞争性比率: 非拒绝值算法, 只要有邻居, 就必须匹配一个在线节点; 普通在线算法, 可以拒绝在线节点, 以增进公平性。 对于长期公平性, 我们分析两种在线算法( 测试和集中) 将算法作为整个系统之间的完全性优化,, 我们通过外部供应/ 显示整个双向交易的对比, 显示所有 的 的 的 的 度, 度, 不要求 展示的 显示的 显示的 的 的 对比性 显示, 我们的 的 显示的 显示的 显示的 的 的 的 的 的 显示的 的 的 显示的 显示的 的 的 的 选择性 的 的 的 选择性 的 的 选择性 的 的 的 选择性 的 。