We investigate online maximum cardinality matching, a central problem in ad allocation. In this problem, users are revealed sequentially, and each new user can be paired with any previously unmatched campaign that it is compatible with. Despite the limited theoretical guarantees, the greedy algorithm, which matches incoming users with any available campaign, exhibits outstanding performance in practice. Some theoretical support for this practical success was established in specific classes of graphs, where the connections between different vertices lack strong correlations - an assumption not always valid. To bridge this gap, we focus on the following model: both users and campaigns are represented as points uniformly distributed in the interval $[0,1]$, and a user is eligible to be paired with a campaign if they are similar enough, i.e. the distance between their respective points is less than $c/N$, with $c>0$ a model parameter. As a benchmark, we determine the size of the optimal offline matching in these bipartite random geometric graphs. In the online setting and investigate the number of matches made by the online algorithm closest, which greedily pairs incoming points with their nearest available neighbors. We demonstrate that the algorithm's performance can be compared to its fluid limit, which is characterized as the solution to a specific partial differential equation (PDE). From this PDE solution, we can compute the competitive ratio of closest, and our computations reveal that it remains significantly better than its worst-case guarantee. This model turns out to be related to the online minimum cost matching problem, and we can extend the results to refine certain findings in that area of research. Specifically, we determine the exact asymptotic cost of closest in the $\epsilon$-excess regime, providing a more accurate estimate than the previously known loose upper bound.

翻译：我们研究在线最大基数匹配问题，这是广告分配中的一个核心问题。在该问题中，用户按顺序出现，每个新用户可以与任何之前未匹配且与其兼容的广告活动配对。尽管理论保障有限，但贪心算法（将新用户与任何可用广告活动匹配）在实践中表现出卓越性能。针对这种实际成功的一些理论支持已在特定图类中建立，但这些图类中不同顶点之间的联系缺乏强相关性——这一假设并非总是成立。为弥补这一差距，我们聚焦于以下模型：用户和广告活动均表示为均匀分布在区间 $[0,1]$ 上的点，当用户与广告活动足够相似时（即各自点之间的距离小于 $c/N$，其中 $c>0$ 为模型参数），该用户才有资格与广告活动配对。作为基准，我们确定了这些二分随机几何图中最优离线匹配的规模。在在线设置中，我们研究了在线算法closest的匹配数量，该算法贪心地将新点与最近的可用邻居配对。我们证明，该算法的性能可与其流体极限进行比较，该极限被刻画为特定偏微分方程的解。通过该PDE解，我们可以计算closest的竞争比，且计算结果表明其竞争比显著优于最坏情况下的理论保证。该模型与在线最小成本匹配问题相关，我们可将结果扩展以完善该领域研究的某些结论。具体而言，我们确定了在$\epsilon$-超额机制下closest的精确渐近成本，提供了比先前已知松散上界更精确的估计。