A Threshold Phenomenon for the Shortest Lattice Vector Problem in the Infinity Norm
This addresses a fundamental computational geometry problem for lattice theory and optimization, providing new algorithmic insights and structural results.
The paper tackles the NP-hard problem of finding the shortest vector in a lattice under the infinity norm, showing it is fixed-parameter tractable with respect to the largest absolute determinant submatrix value, and reveals a threshold phenomenon where beyond a dimension determined by this parameter, the shortest vector has norm one.
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 $Î$, 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 $Î$, 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 $Î$.