Diameters of Permutation Groups on Graphs, Multi-agent Path Planning, and Integer Multiflow on Directed Acyclic Graphs

Let $G$ be an $n$-vertex connected, undirected, simple graph. The vertices of $G$ are populated with $n$ uniquely numbered agents, one on each vertex. Allowing agents on cycles of $G$ to rotate (synchronous rotations along multiple disjoint cycles are allowed), the resulting agent permutations form a group $\G$. Let $diam(\G)$ (the diameter of $\G$) denote the length of the longest product of (cycle) generators required to reach an element of $G$, we show that $diam(\G) = O(n^2)$. Given any initial and goal (agent) configurations, the algorithm from our constructive proofs computes in $O(n^2)$ time whether the goal configuration is reachable and reports such a generator sequence if this is the case. Combined with the results from Kornhauser, Miller, and Spirakis \cite{KorMilSpi84}, we obtain a complete and efficient algorithm for solving a very general version of the multi-agent path planning problem on graphs. The above result is strengthened as we continue to establish that: 1. There exist graphs of which the diameters are at least $\Omega(n \log n)$. 2. Finding time optimal solutions (as measured by the number of legal moves including rotations) to the multi-agent path planning problem is NP-hard. As an added bonus, the latter allows us to show that the maximal integer multiflow problem is NP-hard on directed acyclic graphs, even when edge capacities and supplies (demands) on sources (sinks) all have unit magnitude.