Pebble Motion on Graphs with Rotations: Efficient Feasibility Tests and Planning Algorithms

TL;DR

This paper investigates pebble motion planning on graphs with rotations, providing efficient algorithms for feasibility testing and path planning by analyzing the permutation group diameter induced by the graph.

Contribution

It introduces a novel approach linking pebble motion feasibility to permutation group diameters, enabling linear-time feasibility tests and cubic-time planning algorithms.

Findings

Feasibility is determined by the permutation group diameter.

The permutation group diameter is bounded by O(n^2).

Algorithms achieve linear and cubic time complexity for feasibility and planning.

Abstract

We study the problem of planning paths for $p$ distinguishable pebbles (robots) residing on the vertices of an $n$-vertex connected graph with $p \le n$. A pebble may move from a vertex to an adjacent one in a time step provided that it does not collide with other pebbles. When $p = n$, the only collision free moves are synchronous rotations of pebbles on disjoint cycles of the graph. We show that the feasibility of such problems is intrinsically determined by the diameter of a (unique) permutation group induced by the underlying graph. Roughly speaking, the diameter of a group $\mathbf G$ is the minimum length of the generator product required to reach an arbitrary element of $\mathbf G$ from the identity element. Through bounding the diameter of this associated permutation group, which assumes a maximum value of $O(n^2)$, we establish a linear time algorithm for deciding the feasibility of such problems and an $O(n^3)$ algorithm for planning complete paths.