Metrical Service Systems: Deterministic Strategies

In this paper we introduce a new model of on-line problems called metrical service systems. An instance of a metrical service system is speci ed by a metric space M and a set R of requests, where each r 2 R is a subset of M . There is one server that can move in M . At every time step a request r 2 R is speci ed, and the server must move to a point x 2 r to serve the request r. We also have the following on-line restriction: the choice of x must be made before future requests are given. The sequence of points occupied by the server during this process is called a service. The goal is to minimize the service cost, which is the total distance moved by the server. Even for simple metrical service systems computing the optimal service on-line is impossible. Therefore, we concentrate on so-called competitive strategies that compute a service that only approximates an optimal one. More precisely, a strategy is called c-competitive if, for each request sequence, it computes a service whose cost is bounded by c times the optimal service cost of this sequence plus a constant. First, we develop a theory of on-line strategies for metrical service systems. In particular, we show that the existence of an on-line c-competitive strategy is equivalent to the existence of a c-potential function. The concept of the minimum c-potential is particularly useful in lower-bound proofs. We illustrate our method using two examples. We consider the k-point request problem, a metrical service system where each request set has at most k points. We give optimally competitive deterministic strategies for two cases: (1) for arbitrary k and uniform M , and (2) for k = 2 and arbitrary M . These solutions, and especially the lower bound proofs, illustrate the usefulness of the techniques described in the rst part of the paper. 0.