Intractability of Optimal Multi-Robot Path Planning on Planar Graphs

We study the computational complexity of optimally solving multi-robot path planning problems on planar graphs. For several common time- and distance-based objectives, we show that the associated path optimization problems for multiple robots are all NP-hard on planar graphs. Establishing the computational intractability of optimal multi-robot path planning problems on planar graphs has important practical implications. In particular, from a theoretical perspective, our result implies that the best approach toward solving such problems, when the number of robots is large, is to augment the planar, discrete environment so as to reduce the sharing of paths among robots traveling in opposite directions along those paths. Indeed, such efficiency boosting structures, such as two-way roads and elevated intersections, are rather prevalent in the real world.