Minimizing Travel Time and Latency in Multi-Capacity Ride-Sharing Problems

Motivated by applications in ride-sharing and truck-delivery, we study the problem of matching a number of requests and assigning them to cars. A number of cars are given, each of which consists of a location and a speed, and a number of requests are given, each of which consists of a pick-up location and a drop-off location. Serving a request means that a car must first visit the pick-up location of the request and then visit the drop-off location. Each car can only serve at most c requests. Each assignment can yield multiple different serving routes and corresponding serving times, and our goal was to serve the maximum number of requests with the minimum travel time (called CSsum) and to serve the maximum number of requests with the minimum total latency (called CSlat). In addition, we studied the special case where the pick-up and drop-off locations of a request coincide. Both problems CSsum and CSlat are APX-hard when c & GE;2. We propose an algorithm, called the transportation algorithm (TA), which is a (2c-1)-approximation (resp. c-approximation) algorithm for CSsum (resp. CSlat); these bounds are shown to be tight. We also considered the special case where each car serves exactly two requests, i.e., c=2. In addition to the TA, we investigated another algorithm, called the match-and-assign algorithm (MA). Moreover, we call the algorithm that outputs the best of the two solutions found by the TA and MA the CA. We show that the CA is a two-approximation (resp. 5/3) for CSsum (resp. CSlat), and these ratios are better than the ratios of the individual algorithms, the TA and MA.