Multi-agent Path Planning and Network Flow

TL;DR

This paper establishes a connection between multi-agent path planning and network flow problems, enabling the use of combinatorial algorithms to efficiently find feasible and optimal solutions on graphs.

Contribution

It introduces a reduction of multi-agent path planning to network flow, providing algorithms for feasible, time, and distance optimal solutions with proven bounds.

Findings

Feasible solution paths exist within $n + V - 1$ steps for permutation-invariant goals.

An $O(nVE)$ time algorithm finds such solutions.

Solutions exhibit Pareto optimality for time and distance objectives.

Abstract

This paper connects multi-agent path planning on graphs (roadmaps) to network flow problems, showing that the former can be reduced to the latter, therefore enabling the application of combinatorial network flow algorithms, as well as general linear program techniques, to multi-agent path planning problems on graphs. Exploiting this connection, we show that when the goals are permutation invariant, the problem always has a feasible solution path set with a longest finish time of no more than $n + V - 1$ steps, in which $n$ is the number of agents and $V$ is the number of vertices of the underlying graph. We then give a complete algorithm that finds such a solution in $O(nVE)$ time, with $E$ being the number of edges of the graph. Taking a further step, we study time and distance optimality of the feasible solutions, show that they have a pairwise Pareto optimal structure, and again provide efficient algorithms for optimizing two of these practical objectives.