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In dial-a-ride problems, items have to be transported from a source to a destination. The characteristics of the servers involved as well as the specific requirements of the rides may vary. Problems are defined on some metric space, and the goal is to find a feasible solution that minimizes a certain objective function. The structure of these problems allows for a notation similar to the standard notation for scheduling and queueing problems. We introduce such a notation and show how a class of 7,930 dial-a-ride problem types arises from this approach. In examining their computational complexity, we define a partial ordering on the problem class and incorporate it in the computer program DaRClass. As input DaRClass uses lists of problems whose complexity is known. The output is a classification of all problems into one of three complexity classes: solvable in polynomial time, NP-hard, or open. For a selection of the problems that form the input for DaRClass, we exhibit a proof of polynomial-time solvability or NP-hardness.(Vehicle Routing; Dial-a-Ride; Computational Complexity)

Abstract. In the online dial-a-ride problem (OlDarp), objects must be transported by a server between points in a metric space. Transportation requests ("rides") arrive online, specifying the objects to be transported and the corresponding source and destination. We investigate the OlDarp for the objective of minimizing the maximum flow time. It has been well known that there can be no strictly competitive online algorithm for this objective and no competitive algorithm at all on unbounded metric spaces. However, the question whether on metric spaces with bounded diameter there are competitive algorithms if one allows an additive constant in the definition competitive ratio, had been open for quite a while. We provide a negative answer to this question already on the uniform metric space with three points. Our negative result is complemented by a strictly 2-competitive algorithm for the Online Traveling Salesman Problem on the uniform metric space, a special case of the problem.

Abstract. We consider the general on-line two server problem in which at each step both servers receive a request, which is a point in a metric space. One of the servers has to be moved to its request. The special case where the requests are points on the real line is known as the CNN-problem. It has been a well-known open question if an algorithm with a constant competitive ratio exists for this problem. We answer this question in the affirmative sense by providing the first constant competitive algorithm for the general two-server problem on any metric space.

In on-line dial-a-ride problems servers are traveling in some metric space to serve requests for rides which are presented over time. Each ride is characterized by two points in the metric space, a source, the starting point of the ride, and a destination, the endpoint of the ride. Usually it is assumed that at the release of a request, complete information about the ride is known. We diverge from this by assuming that at the release of a ride, only information about the source is given. At visiting the source, the information about the destination will be made available to the servers. For many practical problems, our model is closer to reality. However, we feel that the lack of information is often a choice, rather than inherent to the problem: additional information can be obtained, but this requires investments in information systems. In this paper we give mathematical evidence that for the problem under study it pays to invest. Prelude.In dial-a-ride problems servers are traveling in some metric space to serve requests for rides. Each ride is characterized by two points in the metric space, a source, the starting point of the ride, and a destination, the endpoint of the ride. The problem is to design routes for the servers through the metric space, such that all requested rides are made and some optimality criterion is met.Dial-a-ride problems have been studied extensively in the literature of operations research, management science, and combinatorial optimization. Traditionally, such combinatorial optimization problems are studied under the assumption that the input of the problem is known completely to the optimizer.In a natural setting of dial-a-ride problems requests for rides are presented over time while the servers are enroute serving other rides, making the problem an on-line optimization problem. Examples in practice are taxi and minibus services, courier services, and elevators. In their on-line setting dial-a-ride problems have been studied in [1] and [4], where single-server versions of the problem are studied, as we will do here. These papers study the problem in which rides are specified completely upon presentation, i.e., both source and destination of the ride become known at the same time. We diverge from this setting here.