Open problem on risk-aware planning in the plane

TL;DR

This paper investigates the computational complexity of risk-aware path planning in the plane, focusing on whether the problem is inherently hard or admits efficient algorithms, especially when considering risk zones and penalties.

Contribution

It formalizes a risk-aware planning problem with a novel cost function balancing path length and risk exposure, and explores its computational complexity.

Findings

The problem reduces to known hard problems in certain cases.

Efficient algorithms exist for minimal-length paths in 2D without risk zones.

Open questions remain about the problem's overall computational complexity.

Abstract

We consider the problem of planning a collision-free path of a robot in the presence of risk zones. The robot is allowed to travel in these zones but is penalized in a super-linear fashion for consecutive accumulative time spent there. We recently suggested a natural cost function that balances path length and risk-exposure time. When no risk zones exists, our problem resorts to computing minimal-length paths which is known to be computationally hard in the number of dimensions. It is well known that in two-dimensions computing minimal-length paths can be done efficiently. Thus, a natural question we pose is "Is our problem computationally hard or not?" If the problem is hard, we wish to find an approximation algorithm to compute a near-optimal path. If not, then a polynomial-time algorithm should be found.