Dynamic traveling repair problem with an arbitrary time window

We consider the online Dynamic Traveling Repair Problem (DTRP) with an arbitrary size time window. In this problem we receive a sequence of requests for service at nodes in a metric space and a time window for each request. The goal is to maximize the number of requests served during their time window. The time to traverse between two points is equal to the distance. Serving a request requires unit time. Irani et al., SODA 2002 considered the special case of a fixed size time window. In contrast, we consider the general case of an arbitrary size time window. We characterize the competitive ratio for each metric space separately. The competitive ratio depends on the relation between the minimum laxity (the minimum length of a time window) and the diameter of the metric space. Specifically, there exists a constant competitive algorithm only when the laxity is larger than the diameter. In addition, we characterize the rate of convergence of the competitive ratio, which approaches 1, as the laxity increases. Specifically, we provide matching lower and upper bounds. These bounds depend on the ratio between the laxity and the optimal TSP solution of the metric space (the minimum distance to traverse all nodes). An application of our result improves the previously known lower bound for colored packets with transition costs and matches the known upper bound. In proving our lower bounds we use an embedding with some special properties.