On the lattice programming gap of the group problems

TL;DR

This paper proves that calculating the lattice programming gap for group problems is NP-hard when the dimension varies and provides bounds based on the cost vector and lattice determinant.

Contribution

It establishes the NP-hardness of computing the lattice programming gap and derives bounds related to the problem's parameters.

Findings

Computing the lattice programming gap is NP-hard for variable dimensions.

Provides lower and upper bounds for the gap based on problem parameters.

Highlights computational complexity in lattice-based group problems.

Abstract

We show that computing the lattice programming gap of the group problems is NP-hard when the dimension is a part of input. We also obtain lower and upper bounds for the gap in terms of the cost vector and the determinant of the lattice.