Maximum Matching on Trees in the Online Preemptive and the Incremental Dynamic Graph Models

We study the MaximumCardinality Matching (MCM) and the MaximumWeight Matching (MWM) problems, on trees and on some special classes of graphs, in the Online Preemptive and the Incremental Dynamic Graph models. In the Online Preemptive model, the edges of a graph are revealed one by one and the algorithm is required to always maintain a valid matching. On seeing an edge, the algorithm has to either accept or reject the edge. If accepted, then the adjacent edges are discarded, and all rejections are permanent. In this model, the complexity of the problems is settled for deterministic algorithms [11, 15]. Epstein et al. [5] gave a 5.356-competitive randomized algorithm for MWM, and also proved a lower bound of 1.693 for MCM. The same lower bound applies for MWM. In an Incremental Dynamic Graph, at each update step an edge is added to the graph. Gupta [7] proved that for any ǫ ≤ 1/2, there exists an algorithm that maintains a (1+ ǫ)-approximate MCM for an incremental bipartite graph in an “amortized”O (