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. [Information Processing Letters, 2013], whereas the special case $k=2$ was studied by Boyar et al. [WADS 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 $3/2$.
Highlights
In the standard framework of online computation, the input to the algorithm is revealed incrementally, i.e., request by request
The algorithm may not alter any previously made decisions while considering later requests. This rather stringent constraint is meant to capture what informally can be described as “the past cannot be undone”; significantly, it is at the heart of adversarial arguments that can be used to argue that the competitive ratio of a given online problem cannot be improved beyond a certain bound
Different approaches to this objective have been considered. One such approach has studied the minimum total re-optimization cost required in order to maintain an optimal solution, see Bernstein et al [2]. Another approach has focused on the best achievable competitive ratio when there is some bound on the allowed re-optimization, which has been first studied by Avitabile et al [1], and is the main model we consider in this paper
Summary
In the standard framework of online computation, the input to the algorithm is revealed incrementally, i.e., request by request. This setting is motivated by similar models that have been studied in the context of the online Steiner tree problem [9] We call this problem the online maximum matching problem under the edge arrival/departure model with edge-bounded recourse. That the known lower bounds for edge-bounded recourse problems in [1, 3] are likewise expressed in terms of the strict competitive ratio This is due, perhaps, to difficulties in applying techniques that extend the lower bounds to the standard definition of the competitive ratio that are inherent to the recourse setting, and which do not arise in the traditional online framework of irrevocable decisions. We will refer to the strict competitive ratio as the “competitive ratio”
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