Hardness of Motion Planning with Obstacle Uncertainty in Two Dimensions

Abstract

We consider the problem of motion planning in the presence of uncertain obstacles, modeled as polytopes with Gaussian-distributed faces (PGDFs). A number of practical algorithms exist for motion planning in the presence of known obstacles by constructing a graph in configuration space, then efficiently searching the graph to find a collision-free path. We show that such an exact algorithm is unlikely to be practical in the domain with uncertain obstacles. In particular, we show that safe 2D motion planning among PGDF obstacles is $\mathrm{NP}$ -hard with respect to the number of obstacles, and remains $\mathrm{NP}$ -hard after being restricted to a graph. Our reduction is based on a path encoding of MAXQHORNSAT and uses the risk of collision with an obstacle to encode variable assignments and literal satisfactions. This implies that, unlike in the known case, planning under uncertainty is hard, even when given a graph containing the solution. We further show by reduction from $3$ -SAT that both safe 3D motion planning among PGDF obstacles and the related minimum constraint removal problem remain $\mathrm{NP}$ -hard even when restricted to cases where each obstacle overlaps with at most a constant number of other obstacles.