In this paper, we address the problem of scheduling a set of robots to complete tasks in a laboratory environment, modelled as a graph, while avoiding collisions. We analyze the dynamic programming algorithm (PA) introduced in arXiv:2402.12019 and present three baselines for comparison: an integer programming approach (IP) that always yields an optimal solution, a greedy algorithm (GA), and a simple randomized algorithm (RA). We show that for a path graph, PA, GA, and RA find solutions several orders of magnitude faster than IP (the optimal baseline), with PA returning optimal results in the vast majority of cases. Our scaled experiments comparing non-optimal algorithms show that the average schedule timespan produced by PA is less than half that of RA and GA. This outperformance is consistent across varying path lengths, task durations and distributions, number and allocations of tasks and robots, and task-to-robot ratios. This work serves two purposes. First, we present three algorithms for scheduling on line graphs, including a novel integer programming formulation for finding optimal solutions. Second, we demonstrate that PA produces near-optimal schedules that outperform all non-optimal baselines while maintaining a comparable runtime. Code is available at https://github.com/sea26-robots/code.

翻译：本文研究了在实验室环境（建模为图）中调度一组机器人完成任务同时避免碰撞的问题。我们分析了arXiv:2402.12019中提出的动态规划算法（PA），并建立了三个比较基准：始终给出最优解的整数规划方法（IP）、贪心算法（GA）以及简单随机算法（RA）。研究表明，在路径图场景下，PA、GA和RA的求解速度比最优基准IP快数个数量级，且PA在绝大多数情况下能返回最优结果。我们通过扩展实验比较非最优算法发现，PA生成的平均调度时间跨度不足RA和GA的一半。这一优势在不同路径长度、任务持续时间与分布、任务与机器人的数量及分配方式、以及任务-机器人比例下均保持一致。本工作具有双重目的：首先，我们提出了三种适用于线图调度的算法，包括一种用于寻找最优解的新型整数规划模型；其次，我们证明PA能在保持相近运行时间的同时，生成优于所有非最优基准的近似最优调度方案。代码发布于https://github.com/sea26-robots/code。