Fast algorithms for optimal multi-robot path planning are sought after in real-world applications. Known methods, however, generally do not simultaneously guarantee good solution optimality and good (e.g., polynomial) running time. In this work, we develop a first low-polynomial running time algorithm, called SplitAndGroup (SaG), that solves the multi-robot path planning problem on grids and grid-like environments, and produces constant factor makespan optimal solutions on average over all problem instances. That is, SaG is an average case O(1)-approximation algorithm and computes solutions with sub-linear makespan. SaG is capable of handling cases when the density of robots is extremely high - in a graph-theoretic setting, the algorithm supports cases where all vertices of the underlying graph are occupied. SaG attains its desirable properties through a careful combination of a novel divide-and-conquer technique, which we denote as global decoupling, and network flow based methods for routing the robots. Solutions from SaG, in a weaker sense, are also a constant factor approximation on total distance optimality.

中文翻译：

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连通环境中的平均工况恒定因子时间和距离最佳多机器人路径规划

在实际应用中，人们寻求用于优化多机器人路径规划的快速算法。然而，已知方法通常不能同时保证良好的解最优性和良好的（例如多项式）运行时间。在这项工作中，我们开发了第一个低多项式运行时间算法，称为SplitAndGroup（SaG），该算法解决了网格和类似网格环境下的多机器人路径规划问题，并针对所有问题平均产生了恒定因子makepan最优解。实例。也就是说，SaG是平均情况O（1）近似算法，并使用亚线性makepan计算解。凹陷能够处理机器人密度非常高的情况-在图论设置中，该算法支持基础图的所有顶点均被占用的情况。SaG通过将新颖的分治技术（我们称为全局解耦）和基于网络流的机器人路由方法进行精心组合，从而获得了理想的性能。从较弱的意义上讲，SaG的解决方案也是总距离最优性的恒定因子近似值。