The Bike Sharing Problem

TL;DR

This paper introduces polynomial-time algorithms for optimizing the travel schedule of autonomous agents and bikes on a line, focusing on reaching the end as quickly as possible and minimizing arrival time with limited bike abandonment.

Contribution

It provides the first polynomial-time algorithms for the Bike Sharing problem and its variant with limited bike abandonment.

Findings

Polynomial-time algorithm for the extsc{Bike Sharing} problem achieving optimal arrival times.

Optimal solution for the extsc{Rearranged Bike Sharing} problem when at most one bike can be abandoned.

Efficient scheduling strategies for cooperative agent-bike transportation on a line.

Abstract

Assume that $m \geq 1$ autonomous mobile agents and $0 \leq b \leq m$ single-agent transportation devices (called {\em 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 \BS 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 \RBS 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 \BS problem that creates an arrival-time optimal schedule for travellers and bikes to travel across the interval. For the \RBS problem, we give an algorithm that gives an optimal solution for the case when at most one of the bikes can be abandoned.