The Bike Sharing Problem

摘要

Assume that m ≥ 1 autonomous mobile agents and 0 ≤ b ≤ m single-agent transportation devices (called bikes) are initially placed at the left endpoint 0 of the unit interval [0, 1]. The agents are identical in capability and can move at speed one. The bikes cannot move on their own, but any agent riding bike i can move at speed v~i ＞ 1. An agent may ride at most one bike at a time. The agents can cooperate by sharing the bikes; an agent can ride a bike for a time, then drop it to be used by another agent, and possibly switch to a different bike. We study two problems. In the Bike Sharing problem, we require all agents and bikes starting at the left endpoint of the interval to reach the end of the interval as soon as possible. In the Relaxed Bike Sharing problem, we aim to minimize the arrival time of the agents; the bikes can be used to increase the average speed of the agents, but are not required to reach the end of the interval. Our main result is the construction of a polynomial time algorithm for the Bike Sharing problem that creates an arrival-time optimal schedule for travellers and bikes to travel across the interval. For the Relaxed Bike Sharing problem, we give an algorithm that gives an optimal solution for the case when at most one of the bikes can be abandoned.