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

TL;DR

This paper addresses the shortest lattice vector problem in the infinity norm for $\Delta$-modular lattices $A\mathbb{Z}^n$, establishing a threshold phenomenon: when the dimension $n$ grows beyond a parameterized bound $f(\Delta)$, the shortest vector must have unit infinity-norm. The authors introduce the function $f(\Delta)$, prove finite bounds $\Delta-1 \le f(\Delta) \le \lceil(\Delta-1)/2\rceil(\Delta-1)$, and derive a fixed-parameter tractable algorithm in $\Delta$ with runtime $\mathcal{O}(m n^2 \Delta^3)$; they further define $F(\Delta)=\max_{i\le \Delta} f(i)$ to obtain polyhedral consequences, notably that vertices of the integer hull lie on faces of dimension at most $F(\Delta)$. Beyond SVP, the work yields sparsity bounds for integer-optimal solutions in standard-form problems and establishes tight lower bounds via totally $\Delta$-modular constructions. Overall, the results illuminate when SVP becomes tractable for Δ-modular lattices and connect lattice structure to polyhedral sparsity in integer programming.

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 $Δ$, 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 $Δ$, 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 $Δ$.

Theorems & Definitions (15)

• Definition 1
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Proposition 1
• Lemma 1
• proof
• Corollary 1
• proof : Proof of Theorem \ref{['thm_shortest_vector']}