Approximation algorithms for car-sharing problems

We consider several variants of a car-sharing problem. Given are a number of requests each consisting of a pick-up location and a drop-off location, a number of cars, and nonnegative, symmetric travel times that satisfy the triangle inequality. Each request needs to be served by a car, which means that a car must first visit the pick-up location of the request, and then visit the drop-off location of the request. Each car can serve two requests. One problem is to serve all requests with the minimum total travel time (called $\CS_{sum}$), and the other problem is to serve all requests with the minimum total latency (called $\CS_{lat}$). We also study the special case where the pick-up and drop-off location of a request coincide. We propose two basic algorithms, called the match and assign algorithm and the transportation algorithm. We show that the best of the resulting two solutions is a $ 2$-approximation for $\CS_{sum}$ (and a $7/5$-approximation for its special case), and a $5/3 $-approximation for $\CS_{lat}$ (and a $3/2$-approximation for its special case); these ratios are better than the ratios of the individual algorithms. Finally, we indicate how our algorithms can be applied to more general settings where each car can serve more than two requests, or where cars have distinct speeds.