We consider the online version of the maximum vertex disjoint path problem when the underlying network is a tree. In this problem, a sequence of requests arrives in an online fashion, where every request is a path in the tree. The online algorithm may accept a request only if it does not share a vertex with a previously accepted request. The goal is to maximize the number of accepted requests. It is known that no online algorithm can have a competitive ratio better than Ω(log n) for this problem, even if the algorithm is randomized and the tree is simply a line. Obviously, it is desirable to beat the logarithmic lower bound. Adler and Azar [SODA 1999] showed that if preemption is allowed (namely, previously accepted requests may be discarded, but once a request is discarded it can no longer be accepted), then there is a randomized online algorithm that achieves constant competitive ratio on the line. In the current work we present a randomized online algorithm with preemption that has constant competitive ratio on any tree. Our results carry over to the related problem of maximizing the number of accepted paths subject to a capacity constraint on vertices (in the disjoint path problem this capacity is 1). Moreover, if the available capacity is at least 4, randomization is not needed and our online algorithm becomes deterministic.
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