We study the online maximum matching problem in a model in which the edges are associated with a known recourse parameter k. An online algorithm for this problem has to maintain a valid matching while edges of the underlying graph are presented one after the other. At any moment the algorithm can decide to include an edge into the matching or to exclude it, under the restriction that at most k such actions per edge take place, where k is typically a small constant. This problem was introduced and studied in the context of general online packing problems with recourse by Avitabile et al. (Inf Process Lett 113(3):81–86, 2013), whereas the special case \(k=2\) was studied by Boyar et al. (Proceedings of the 15th workshop on algorithms and data structures (WADS), pp 217–228, 2017). In the first part of this paper we consider the edge arrival model, in which an arriving edge never disappears from the graph. Here, we first show an improved analysis on the performance of the algorithm AMP of Avitabile et al., by exploiting the structure of the matching problem. In addition, we show that the greedy algorithm has competitive ratio 3/2 for every even k and ratio 2 for every odd k. Moreover, we present and analyze an improvement of the greedy algorithm which we call L-Greedy, and we show that for small values of k it outperforms the algorithm AMP. In terms of lower bounds, we show that no deterministic algorithm better than \(1+1/(k-1)\) exists, improving upon the known lower bound of \(1+1/k\). The second part of the paper is devoted to the edge arrival/departure model, which is the fully dynamic variant of online matching with recourse. The analysis of L-Greedy and AMP carry through in this model; moreover we show a lower bound of \((k^2-3k+6) / (k^2-4k+7)\) for all even \(k \ge 4\). For \(k\in \{2,3\}\), the competitive ratio is 3/2.

我们在模型的边缘与已知资源参数k相关的模型中研究在线最大匹配问题。针对此问题的在线算法必须保持有效的匹配，而基础图的边缘则一个接一个地呈现。在每个边缘最多发生k个此类动作的限制下，算法可以随时决定将边缘包括在匹配中或排除在匹配之外，其中k通常是一个较小的常数。这个问题是在Avitabile等人利用资源求助的一般在线包装问题的背景下引入和研究的。（Inf Process Lett 113（3）：81–86，2013），而特殊情况\（k = 2 \）由Boyar等人研究。（第15届算法和数据结构研讨会（WADS）的会议记录，2017年，第217-228页）。在本文的第一部分中，我们考虑边缘到达模型，其中到达的边缘永远不会从图中消失。在这里，我们首先通过利用匹配问题的结构，对Avitabile等人的算法AMP的性能进行了改进的分析。此外，我们表明，贪心算法的竞争比为每偶数k 3/2 ，每奇数k 2。此外，我们提出并分析了称为L - Greedy的贪婪算法的改进，并证明了对于k的较小值它的性能优于算法AMP。就下界而言，我们证明不存在比\（1 + 1 /（k-1）\）更好的确定性算法，并改进了已知的\（1 + 1 / k \）下界。本文的第二部分专门介绍了边缘到达/离开模型，它是具有追索权的在线匹配的全动态变体。该模型对L - Greedy和AMP进行了分析；此外，对于所有偶数（{k \ ge 4 \），我们显示\（（k ^ 2-3k + 6）/（k ^ 2-4k + 7）\）的下限。对于\（k \ in \ {2,3 \} \），竞争比是3/2。