On the Complexity of the Orbit Problem

TL;DR

This paper investigates the computational complexity of the Orbit Problem in higher dimensions, establishing polynomial-time solvability for one-dimensional cases and NP^RP complexity for dimensions two and three.

Contribution

It resolves two longstanding questions by characterizing the complexity of the Orbit Problem for specific dimensions, extending prior understanding.

Findings

Orbit Problem is polynomial-time solvable in 1D.

Orbit Problem is in NP^RP for 2D and 3D.

Provides complexity classifications for higher-dimensional cases.

Abstract

We consider higher-dimensional versions of Kannan and Lipton's Orbit Problem---determining whether a target vector space V 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 V has dimension one, this problem is solvable in polynomial time, and when V has dimension two or three, the problem is in NP^{RP}.