Approximation algorithms for car-sharing problems

TL;DR

This paper introduces approximation algorithms for various car-sharing problems, achieving improved bounds for minimizing total travel time and latency, and extends solutions to more general scenarios with multiple requests per car and different speeds.

Contribution

It presents new approximation algorithms with proven bounds for car-sharing problems, including special cases and generalizations beyond the basic model.

Findings

2-approximation for total travel time in general case

7/5-approximation for total travel time in special case

5/3-approximation for total latency in general case

Abstract

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.