Online Maximum Matching with Recourse

We study the online maximum matching problem with recourse 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 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. [AMP13], whereas the special case k = 2 has been studied by Boyar et al. [BFKL17]. 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 given in [AMP13], by exploiting the structure of the specific problem. In addition, we extend the result of [BFKL17] and show that the greedy algorithm has ratio 3/2 for every even positive k and ratio 2 for every odd k. Moreover, we present and analyze L-Greedy — an improvement of the greedy algorithm — which for small values of k outperforms the algorithm of [AMP13]. In terms of lower bounds, we show that no deterministic algorithm better than 1 + 1/(k − 1) exists, improving upon the lower bound of 1 + 1/k shown in [AMP13]. The second part of the paper is devoted to the edge arrival/departure model, which is the fully dynamic variant of the online matching problem with recourse. The analysis of L-Greedy carries through in this model; moreover we show a general lower bound of (k − 3k + 3)/(k − 4k + 5) for all even k ≥ 4 and provide the stronger bound of 10/7 for k = 4. For k ∈ {2, 3}, the competitive ratio is 3/2.