Robots are becoming an increasingly common part of scientific work within laboratory environments. In this paper, we investigate the problem of designing \emph{schedules} for completing a set of tasks at fixed locations with multiple robots in a laboratory. We represent the laboratory as a graph with tasks placed on fixed vertices and robots represented as agents, with the constraint that no two robots may occupy the same vertex at any given timestep. Each schedule is partitioned into a set of timesteps, corresponding to a walk through the graph (allowing for a robot to wait at a vertex to complete a task), with each timestep taking time equal to the time for a robot to move from one vertex to another and each task taking some given number of timesteps during the completion of which a robot must stay at the vertex containing the task. The goal is to determine a set of schedules, with one schedule for each robot, minimising the number of timesteps taken by the schedule taking the greatest number of timesteps within the set of schedules. We show that this problem is NP-complete for many simple classes of graphs, the problem of determining the fastest schedule, defined by the number of time steps required for a robot to visit every vertex in the schedule and complete every task assigned in its assigned schedule. Explicitly, we provide this result for complete graphs, bipartite graphs, star graphs, and planar graphs. Finally, we provide positive results for line graphs, showing that we can find an optimal set of schedules for $k$ robots completing $m$ tasks of equal length of a path of length $n$ in $O(kmn)$ time, and a $k$-approximation when the length of the tasks is unbounded.

翻译：机器人正日益成为实验室环境中科学工作的常见组成部分。本文研究了在实验室中通过多台机器人完成一组固定位置任务的调度设计问题。我们将实验室表示为图，任务放置在固定顶点上，机器人表示为智能体，并约束任意两个机器人不得在同一时间步占据同一顶点。每个调度被划分为一组时间步，对应于在图上的行走（允许机器人停留在顶点以完成任务），每个时间步所需时间等于机器人从一个顶点移动到另一个顶点的时间，而每个任务在完成期间需要机器人停留在任务所在顶点若干时间步。目标是确定一组调度（每台机器人对应一个调度），最小化该组调度中最大时间步数对应的调度所花费的时间步数。我们证明该问题对于许多简单图类（包括完全图、二分图、星图和平凡图）是NP完全的，即确定最快调度（定义为机器人访问调度中每个顶点并完成其分配任务所需的时间步数）的问题。具体地，我们为完全图、二分图、星图和平凡图提供了这一结果。最后，我们为线图提供了积极结果：对于在长度为n的路径上完成m个等长任务的k台机器人，我们可在O(kmn)时间内找到最优调度集，并在任务长度无界时提供k-近似算法。