Serving Requests with On-line Routing

In this paper we consider the on-line version of the routing problem with release times. Formally, it consists in a metric space M with a distinguished point o (the origin), plus a sequence of triples < ti; pi; ri > where pi is a point of M , ri speciies the rst moment in which the request is ready to be served, and ti represents the moment in which the request is known. A server located at point o at time 0 that moves at constant unit speed must serve the sequence of requests trying to minimize the total time till all the requests are served. An on-line server follows a solution computed on the basis of the set of requests presented in the past; this solution may be updated each time some new piece of information is known. We show that no on-line algorithm, neither deterministic nor randomized, can achieve a competitive factor lower than 2. We give a 5/2-competitive exponential algorithm and a 3-competitive polynomial algorithm for the plane, and a 7=3-competitive algorithm for the line.