Finding an Integral vector in an Unknown Polyhedral Cone

TL;DR

This paper introduces an algorithm to find an integral vector within an unknown polyhedral cone defined by inequalities with bounded integer coefficients, relying only on the existence of a known non-zero integral solution.

Contribution

The paper presents a novel algorithm that locates an integral vector in a polyhedral cone without explicit knowledge of the defining matrix, using only bounds on coefficients and a known solution.

Findings

Successfully finds an integral vector with bounded maximum element.

Operates without explicit knowledge of the cone's defining matrix.

Provides bounds on the maximum element of the found vector.

Abstract

We present an algorithm to find an integral vector in the polyhedral cone $\Gamma=\{X | \textbf{A}X \leq \textbf{0}\}$, without assuming the explicit knowledge of $\textbf{A}$. About the polyhedral cone, $\Gamma$, it is only given that, (i) the elements of \textbf{A} are in $\{-d,-d+1,\...,0,\...,d-1,d\}$, $d \in \mathbb{N}$, and, (ii) $Y=[y(1),y(2),\...,y(n)]$ is a non-zero integral solution to $\Gamma$. The proposed algorithm finds a non-zero integral vector in $\Gamma$ such that its maximum element is less than ${(2d)^{2^{n-1}-1}}/{2^{n-1}}$.