Online maximum matching with recourse
- PDF / 663,357 Bytes
- 34 Pages / 439.37 x 666.142 pts Page_size
- 90 Downloads / 247 Views
Online maximum matching with recourse Spyros Angelopoulos1
· Christoph Dürr1
· Shendan Jin1
© Springer Science+Business Media, LLC, part of Springer Nature 2020
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. (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 LGreedy 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. Keywords Matching · Online algorithms · Competitive analysis · Recourse
Supported by ANR OATA (ANR-15-CE40-0015), DIM RFSI DACM and Labex Mathématique Hadamard. Preliminary version appeared in the Proceedings of the 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS), 2018.
B
Spyros Angelopoulos [email protected]
Extended author information available on the last page of the article
123
Journal of Combinatorial Optimization
1 Introduction In the standard framework of online computation, the input to the algorithm is revealed incrementally, i.e., as a sequence of requests. For each such requested input item, the online algorithm must make a decision that is typically irrevocable, in the sense that the algorithm commits, in a permanent manner, to the decision associated with the request. More precisely, 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
Data Loading...
