A Threshold Phenomenon for the Shortest Lattice Vector Problem in the Infinity Norm

TL;DR

This paper investigates the shortest vector problem in the infinity norm for lattices, revealing a threshold phenomenon where vectors of norm one emerge beyond a certain dimension related to the lattice's determinant properties.

Contribution

It introduces a fixed parameter tractable algorithm based on a structural threshold phenomenon for the shortest vector problem in the infinity norm.

Findings

NP-hardness of the problem in general

Existence of a dimension threshold where shortest vectors have norm one

Applications to integer optimization and polyhedral faces

Abstract

One important question in the theory of lattices is to detect a shortest vector: given a norm and a lattice, what is the smallest norm attained by a non-zero vector contained in the lattice? We focus on the infinity norm and work with lattices of the form $A\mathbb{Z}^n$, where $A$ has integer entries and is of full column rank. Finding a shortest vector is NP-hard. We show that this task is fixed parameter tractable in the parameter $\Delta$, the largest absolute value of the determinant of a full rank submatrix of $A$. The algorithm is based on a structural result that can be interpreted as a threshold phenomenon: whenever the dimension $n$ exceeds a certain value determined only by $\Delta$, then a shortest lattice vector attains an infinity norm value of one. This threshold phenomenon has several applications. In particular, it reveals that integer optimal solutions lie on faces of the given polyhedron whose dimensions are bounded only in terms of $\Delta$.