New research proves sampling-based reachability has unavoidable exponential complexity

Reachability analysis is a cornerstone of safety verification in robotics, control, and neural network verification. Classical methods like Hamilton-Jacobi reachability and set propagation fail in high dimensions due to exponential computational cost. Sampling-based methods have emerged as a scalable alternative, but until now there was no rigorous understanding of how geometry, dynamics, and sampling interact to affect accuracy. A new paper from Jixian Liu, Ihab Tabbara, Hussein Sibai, and Enrique Mallada (arXiv:2607.18606, CS.RO) tackles this by framing reachable-set recovery as a geometric support estimation problem.

The authors first identify two key regularity properties that make recovery well-posed: the initial set's complement must have positive reach (a smoothness condition), and the dynamics must be Lipschitz continuous. When these hold, a probability-mass guarantee can be upgraded to a Hausdorff distance accuracy bound. However, they then prove that even under these ideal conditions, the sample complexity is fundamentally exponential: a minimax lower bound of Ω((e^{LT}/r)^n) samples is required for any estimator, where n is state dimension and T is time horizon. Experiments on nonlinear systems confirm that while adversarial sampling can improve constants, it cannot change the exponential scaling. This work provides definitive limits that system designers must respect when deploying sampling-based verification in high-dimensional safety-critical systems.