Motion planning in dimensions higher than two

Determine the algorithmic status of multi-robot motion planning in fixed-dimensional Euclidean spaces of dimension greater than two.

Background

The paper’s FPTAS framework extends from planar polygonal workspaces and polygonal robots to fixed-dimensional spaces with polytope workspaces and polytope robots. Nevertheless, the authors state that the multi-robot motion-planning problem in dimensions greater than two remains widely unresolved, leaving its broader algorithmic status as an open direction.

References

To the best of our knowledge, the problem in Rd for d > 2 is widely open.

— FPTAS for Polygonal Multi-Robot Motion Planning  (2610.20244 - Bandyapadhyay et al., 14 Oct 2026) in Section 1, Introduction, subsection “Our result”