Two robots need to move from start to finish. The workspace contains protective polygonal regions — safe zones where the robots are shielded from some environmental hazard. The objective: minimize the total time both robots spend outside these safe zones.

This is not shortest-path planning. A longer path that stays inside protective regions can have less exposure than a shorter path that crosses open space. The optimization trades distance against safety, and the trade-off has geometric structure: the optimal paths follow the boundaries of safe regions, hugging their edges before crossing exposed gaps at maximum speed.

The algorithm runs in O(n^4 log n) time for two robots, where n counts the vertices of the protective polygons. The key insight is that the exposure-minimizing paths form a finite combinatorial structure: there are only polynomially many qualitatively distinct routing options, each defined by which polygon edges the robots follow and where they transition between regions.

Safety as a geometric optimization target is different from the usual robotics objectives (shortest path, minimum time, collision avoidance). It introduces a spatial heterogeneity — some locations are safe, others aren’t — that is absent from standard path planning. The result is paths that look irrational from a distance perspective (they meander, double back, cluster near polygon boundaries) but are optimal from an exposure perspective.