We investigate the power of randomized algorithms for the maximum cardinality matching (MCM) and the maximum weight matching (MWM) problems in the online preemptive model. In this 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. The complexity of the problem is settled for deterministic algorithms [6, 8].Almost nothing is known for randomized algorithms. A lower bound of 1.693 is known for MCM with a trivial upper bound of two. An upper bound of 5.356 is known for MWM. We initiate a systematic study of the same in this paper with an aim to isolate and understand the difficulty. We begin with a primal-dual analysis of the deterministic algorithm due to [6]. All deterministic lower bounds are on instances which are trees at every step. For this class of (unweighted) graphs we present a randomized algorithm which is 28 15 -competitive. The analysis is a considerable extension of the (simple) primal-dual analysis for the deterministic case. The key new technique is that the distribution of primal charge to dual variables depends on the "neighborhood" and needs to be done after having seen the entire input. The assignment is asymmetric: in that edges may assign different charges to the two end-points. Also the proof depends on a non-trivial structural statement on the performance of the algorithm on the input tree.The other main result of this paper is an extension of the deterministic lower bound of Varadaraja [8] to a natural class of randomized algorithms which decide whether to accept a new edge or not using independent random choices. This indicates that randomized algorithms will have to use dependent coin tosses to succeed. Indeed, the few known randomized algorithms, even in very restricted models follow this.We also present the best possible 4 3 -competitive randomized algorithm for MCM on paths.

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We study the Maximum Cardinality Matching (MCM) and the Maximum Weight 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 on the competitive ratio of (1 + ln 2) ≈ 1.693 for MCM. The same lower bound applies for MWM.In the Incremental Dynamic Graph model, at each step an edge is added to the graph, and the algorithm is supposed to quickly update its current matching. 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" update time of O log 2 n In this paper we show that some of the results can be improved for trees, and for some special classes of graphs. In the online preemptive model, we present a 64/33-competitive (in expectation) randomized algorithm (which uses only two bits of randomness) for MCM on trees.Inspired by the above mentioned algorithm for MCM, we present the main result of the paper, a randomized algorithm for MCM with a "worst case" update time of O(1), in the incremental dynamic graph model, which is 3/2-approximate (in expectation) on trees, and 1.8-approximate (in expectation) on general graphs with maximum degree 3. Note that this algorithm works only against an oblivious adversary. Hence, we derandomize this algorithm, and give a (3/2 + ǫ)-approximate deterministic algorithm for MCM on trees, with an amortized update time of O(1/ǫ).We also present a minor result for MWM in the online preemptive model, a 3-competitive (in expectation) randomized algorithm (that uses only O(1) bits of randomness) on growing trees (where the input revealed upto any stage is always a tree, i.e. a new edge never connects two disconnected trees).