Feasibility of Motion Planning on Acyclic and Strongly Connected Directed Graphs

TL;DR

This paper investigates the computational feasibility of motion planning on directed graphs, providing linear-time algorithms for acyclic and strongly connected cases, with structural decomposition insights for the latter.

Contribution

It introduces linear-time algorithms for motion planning feasibility on acyclic and strongly connected directed graphs, utilizing a novel structural decomposition for the latter.

Findings

Feasibility can be decided in linear time for acyclic directed graphs.

A structural theorem decomposes strongly connected graphs into strongly biconnected components.

An efficient algorithm is provided for motion planning on strongly connected graphs.

Abstract

Motion planning is a fundamental problem of robotics with applications in many areas of computer science and beyond. Its restriction to graphs has been investigated in the literature for it allows to concentrate on the combinatorial problem abstracting from geometric considerations. In this paper, we consider motion planning over directed graphs, which are of interest for asymmetric communication networks. Directed graphs generalize undirected graphs, while introducing a new source of complexity to the motion planning problem: moves are not reversible. We first consider the class of acyclic directed graphs and show that the feasibility can be solved in time linear in the product of the number of vertices and the number of arcs. We then turn to strongly connected directed graphs. We first prove a structural theorem for decomposing strongly connected directed graphs into strongly biconnected components.Based on the structural decomposition, we give an algorithm for the feasibility of motion planning on strongly connected directed graphs, and show that it can also be decided in time linear in the product of the number of vertices and the number of arcs.