Improved bounds for randomized preemptive online matching

Preemptive online algorithms for the maximum matching problem maintain a valid matching M while edges of the underlying graph are presented one after the other. When presented with an edge e, the algorithm should decide whether to augment the matching M by adding e (in which case e may be removed later on) or to keep M in its current form without adding e (in which case e is lost for good). The objective is to eventually hold a matching M with maximum weight. The main contribution of this paper is to establish new lower and upper bounds on the competitive ratio achievable by randomized preemptive online algorithms: • We provide a lower bound of 1+ln 2 ≈ 1.693 on the competitive ratio of any randomized algorithm for the maximum cardinality matching problem. This result improves on the currently best known bound of e/(e − 1) ≈ 1.581, implied by the work of Karp, Vazirani, and Vazirani [STOC ’90] on a closely-related model with vertex arrivals. • We devise a randomized algorithm that achieves an expected competitive ratio of 5.356 for maximum weight matching. This finding demonstrates the power of randomization in this context, showing how to beat the tight bound of 3 + 2 √ 2 ≈ 5.828 for deterministic algorithms, obtained by combining the 5.828 upper bound of McGregor [APPROX ’05] and the 5.828 lower bound of Varadaraja [ICALP ’11].