Minimum-Link Covering Trails for any Hypercubic Lattice

TL;DR

This paper investigates the minimum link-length of covering trails in high-dimensional hypercubic lattices, showing that previous rectilinear assumptions do not hold in the general NP-complete case and providing bounds on link-length growth.

Contribution

It extends the study of covering paths to non-rectilinear cases, proving the original theorem's limitations and establishing new bounds for minimal link-length in high dimensions.

Findings

Original rectilinear theorem does not hold without axis-parallel constraints.

Minimal link-length grows faster than previously conjectured as dimension increases.

A multiplicative constant c ≥ 1.5 is necessary for lower order term bounds.

Abstract

In 1994, Kranakis et al. published a conjecture about the minimum link-length of every rectilinear covering path for the $k$-dimensional grid $P(n,k) := \{0,1, \dots, n-1\} \times \{0,1, \dots, n-1\} \times \cdots \times \{0,1, \dots, n-1\}$. In this paper, we consider the general, NP-complete, Line-Cover problem, where the edges are not required to be axis-parallel, showing that the original Theorem 1 by Kranakis et al. no longer holds when the aforementioned constraint is disregarded. Furthermore, for any $n$ greater than two, as $k$ approaches infinity, the link-length of any minimal (non-rectilinear) polygonal chain does not exceed Kranakis' conjectured value of $\frac{k}{k-1} \cdot n^{k-1}+O(n^{k-2})$ only if we introduce a multiplicative constant $c \geq 1.5$ for the lower order terms (e.g., if we select $n=3$ and assume that $c<1.5$, starting from a sufficiently large $k$, it is not possible to visit all the nodes of $P(n,k)$ with a trail of link-length $\frac{k}{k-1} \cdot n^{k-1}+c \cdot n^{k-2}$).