Online maximum matching with recourse

Abstract

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. [1], whereas the special case k = 2 was studied by Boyar et al. [3]. 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 given in [1], by exploiting the structure of the matching problem. In addition, we extend the result of [3] and 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 of [1]. 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 [1]. 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 ≥ 4. For k ∈ {2, 3}, the competitive ratio is 3/2. 2012 ACM Subject Classification Mathematics of computing → Graph algorithms, Theory of computation → Online algorithms Keywords and phrases Competitive ratio, maximum cardinality matching, recourse

The Power of Amortized Recourse for Online Graph Problems

“…There is a line of research on online matching problems with recourse. Angelopoulos et al [2] studied a more general setting for Maximum Cardinality Matching and showed that given that no element incurs more than k recourse, there exists an algorithm that attains a competitive ratio of 1 + O(1/ √ k). Megow and Nölke [20] showed that for the Min-Cost Bipartite Matching problem, constant competitiveness is achievable with amortized recourse O(log n), where n is the number of requests.…”

Relative Worst-order Analysis

“…Whereas late rejection has had significant focus, late acceptance was studied in [Boyar et al 2016], followed by a systematic study of all combinations of a number of classic graph problems combined with either late accept, late reject, or both (where late reject is irrevocable) in [Boyar et al 2017a], detailing the performance implications of each choice. In [Angelopoulos et al 2018], the work of [Boyar et al 2017a] was continued for the matching problem, considering more levels of late accept/reject.…”

Online Graph Matching Problems with a Worst-Case Reassignment Budget

“…When an object (vertex or edge) arrives, we find a shortest (if the model is unweighted) or most profitable (if weighted) one among all augmenting paths that contain the object. 2 If such a path does not exist or is longer than k − 1, do nothing; otherwise, augment the matching along the path.…”

“…Although their model significantly differs from our proposed model, we note that their analysis, with hindsight, would have been useful in proving some special cases (k = 4) of Theorems 1 and 5. Angelopoulos, Dürr, and Jin [2] considered the case where the decision can be revoked at most k−1 times and gave a 1…”