The Polyhedron-Hitting Problem

TL;DR

This paper characterizes the decidability of the Polyhedron-Hitting Problem, determining when it can be algorithmically solved based on the dimensions involved, and links some cases to open number-theoretic problems.

Contribution

It provides a complete classification of the decidability landscape for the Polyhedron-Hitting Problem based on ambient and target dimensions.

Findings

Decidability established for certain dimension pairs.

Hardness results linked to open number-theoretic problems.

Complete characterization of the problem's decidability landscape.

Abstract

We consider polyhedral versions of Kannan and Lipton's Orbit Problem (STOC '80 and JACM '86)---determining whether a target polyhedron V may be reached from a starting point x under repeated applications of a linear transformation A in an ambient vector space Q^m. In the context of program verification, very similar reachability questions were also considered and left open by Lee and Yannakakis in (STOC '92). We present what amounts to a complete characterisation of the decidability landscape for the Polyhedron-Hitting Problem, expressed as a function of the dimension m of the ambient space, together with the dimension of the polyhedral target V: more precisely, for each pair of dimensions, we either establish decidability, or show hardness for longstanding number-theoretic open problems.