Bounds for the traveling salesman paths of two-dimensional modular lattices

TL;DR

This paper establishes tight bounds for the traveling salesman path in two-dimensional modular lattices and applies these results to Kronecker point sets, leveraging lattice vector theory and Jacobi-Perron algorithms.

Contribution

It provides the first tight bounds for TSP paths in 2D modular lattices and extends these bounds to Kronecker point sets using advanced lattice and algorithmic techniques.

Findings

Tight upper and lower bounds for TSP paths in 2D modular lattices

Application of bounds to Kronecker point sets

Use of lattice shortest vector theory and Jacobi-Perron algorithms

Abstract

We present tight upper and lower bounds for the traveling salesman path through the points of two-dimensional modular lattices. We use these results to bound the traveling salesman path of two-dimensional Kronecker point sets. Our results rely on earlier work on shortest vectors in lattices as well as on the strong convergence of Jacobi-Perron type algorithms.