On the Complexity of the Orbit Problem

Abstract

We consider higher-dimensional versions of Kannan and Lipton’s Orbit Problem—determining whether a target vector space ν may be reached from a starting point x under repeated applications of a linear transformation A . Answering two questions posed by Kannan and Lipton in the 1980s, we show that when ν has dimension one, this problem is solvable in polynomial time, and when ν has dimension two or three, the problem is in NP RP .