Collision-Free Robot Scheduling

2024-02-19

ALGOWIN

AI Summary

This paper studies the conflict-free scheduling problem for multi-robot coordination in graph-structured laboratory environments, where robots execute fixed-location tasks at vertices and must never occupy the same vertex simultaneously; the objective is to minimize the makespan—the maximum number of time steps required by any robot. We first establish NP-completeness of the problem on complete graphs, bipartite graphs, star graphs, and planar graphs. For linear path graphs, we propose an optimal O(kmn)-time algorithm and a k-approximation algorithm. For line graphs, we devise a polynomial-time exact algorithm: it yields optimal schedules for unit-length tasks and guarantees a k-approximation for variable-length tasks. Our work integrates graph-theoretic modeling, computational complexity analysis, and combinatorial scheduling design, systematically characterizing the problem’s hardness boundary and delivering tight algorithmic results for fundamental graph classes.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Intelligent Robots: Multi-Robot Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

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.

Problem

• Design collision-free schedules for multiple robots in labs
• Minimize maximum timesteps for robots completing tasks on graphs
• Prove NP-completeness for various graph types and provide solutions

Innovation

• Graph-based robot scheduling for task completion
• NP-complete analysis for various graph types
• Optimal scheduling algorithm for line graphs