We study the Maximum Cardinality Matching (MCM) and the Maximum Weight Matching (MWM) problems on trees in the Online Preemptive model, and in the Incremental Dynamic Graph model. In the {\em 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. Epstein et al. 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 a dynamic graph, at each update step an edge is added or deleted from the graph. If only insertions are allowed, then the graph is said to be an {\em Incremental Dynamic Graph}. Gupta proved that for any $\epsilon\leq 1/2$, there exists an algorithm that maintains a $(1+\epsilon )$-approximate MCM for an incremental bipartite graph in an "amortized" $O\left(\frac{\log^2 n}{\epsilon^4}\right)$ update time. #R##N#No better bounds are known even in special cases. In this paper we show that some of the results can be improved in the case of trees. In the online preemptive model, we present a $3$-competitive barely random algorithm (that uses only $O(1)$ bits of randomness) for MWM on growing trees (where the new edge revealed will be incident on some vertex of the already seen tree), and also present a $64/33$-competitive barely random algorithm (which uses only two bits of randomness) for MCM on trees. #R##N#Inspired by the above mentioned algorithm for MCM, we present a $3/2$-approximate (in expectation) randomized algorithm for MCM on trees with a "worst case" update time of $O(1)$, in the incremental dynamic graph model.
['Sumedh Tirodkar', 'Sundar Vishwanathan']
Maximum Matching on Trees in the Online Preemptive and the Incremental Dynamic Graph Models