Abstract We study a variant of the offline Dial-a-Ride problem, where each request has a source and destination and the goal is to maximize the number of requests served within a specified time limit. We investigate this problem for the uniform metric space and show that the problem is NP-hard. We then present a 2/3 approximation algorithm called $$\textsc {twochain}$$ T W O C H A I N , which simply looks for pairs of requests that are “chained” together and serves those before serving requests that are not connected to any others. We also show that a natural generalization of this algorithm, k-chain, has an approximation ratio at most 7/9. We also analyze the longest-chain-first algorithm for the problem, characterizing graphs on which it is optimal, and showing that it has an approximation ratio no better than 5/6. Our experiments on all of these algorithms show that $$\textsc {twochain}$$ T W O C H A I N is a promising algorithm, performing nearly as well as more computationally intensive variants. We dedicate this article to the memory of Gerhard Woeginger, whose life and work greatly influenced our professional lives, as expanded upon in the Acknowledgments. Woeginger’s prolific research in scheduling, matching, bin-packing, TSP, and online algorithms in general, all served as important parts of the foundation on which our own scholarly pursuits were shaped and formed over the years. Woeginger also studied Dial-a-Ride (DARP) Problems, as DARP is a generalization both of scheduling problems and of TSP, which were two of his most active areas of research.

We study a variation of offline Dial-a-Ride, where each request has not only a source and destination, but also a revenue that is earned for serving the request. We investigate this problem for the uniform metric space with uniform revenues. While we present a study on a simplified setting of the problem that has limited practical applications, this work provides the theoretical foundation for analyzing the more general forms of the problem. Since revenues are uniform the problem is equivalent to maximizing the number of served requests. We show that the problem is NP-hard and present a 2/3 approximation algorithm. We also show that a natural generalization of this algorithm has an approximation ratio at most 7/9.

We study a variation of the Online-Dial-a-Ride Problem where each request comes with not only a source, destination and release time, but also has an associated revenue. The server's goal is to maximize its total revenue within a given time limit, T. We show that the competitive ratio is unbounded for any deterministic online algorithm for the problem. We then provide a 3-competitive algorithm for the problem in a uniform metric space and a 6-competitive algorithm for the general case of weighted graphs (under reasonable assumptions about the input instance). We conclude with an experimental evaluation of our algorithm in simulated settings inspired by real-world Dial-a-Ride data. Experimental results show that our algorithm performs well when compared to an offline version of the algorithm and a greedy algorithm.