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.• We devise a randomized algorithm that achieves an expected competitive ratio of 5.356 for maximum weight matching.