Computational complexity of two-robot motion planning in polygonal environments

Determine whether motion planning for two unit-square robots in a general polygonal workspace under the Euclidean metric is NP-hard or polynomial-time solvable for each of the min-sum, min-max, and min-makespan objectives.

Background

The paper distinguishes approximation from exact computational complexity. Although an FPTAS is known for the min-sum objective with two unit-square robots, the exact complexity of the corresponding problems in a general polygonal workspace under the Euclidean metric remains unresolved. The authors explicitly identify both possible outcomes—NP-hardness and polynomial-time solvability—for each of the three cost objectives.

References

In a general polygonal workspace W under the Euclidean metric, however, it is open whether the problem is NP-hard or polynomial-time solvable for any of the cost functions.

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

Finally, it is currently open whether 2D polygonal motion planning with a fixed number of robots is polynomial-time solvable or NP-hard, while the problem in 3D is known to be NP-hard even with one robot [8]. Settling the computational complexity of the 2D problem is an important open question.

— FPTAS for Polygonal Multi-Robot Motion Planning  (2610.20244 - Bandyapadhyay et al., 14 Oct 2026) in Section 8, Conclusion and future work