Improved Bounds for Online Preemptive Matching

When designing a preemptive online algorithm for the maximum matching problem, we wish to 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 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, thus improving on the currently best known bound of e/(e− 1) ≈ 1.581 due to Karp, Vazirani, and Vazirani [STOC’90]. 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 recent 5.828 lower bound of Varadaraja [ICALP’11]. 1998 ACM Subject Classification F.2.2. Nonnumerical algorithms and problems