Computing braid groups of graphs with applications to robot motion planning

Abstract

We design an algorithm writing down presentations of graph braid groups. Generators are represented in terms of actual motions of robots moving without collisions on a given graph. A key ingredient is a new motion planning algorithm whose complexity is linear in the number of edges and quadratic in the number of robots. The computing algorithm implies that 2-point braid groups of all light planar graphs have presentations where all relators are commutators.