An Improved Upper Bound for the Euclidean TSP Constant Using Band Crossovers

TL;DR

This paper refines the upper bound for the Euclidean TSP constant by introducing a band crossover heuristic, supported by simulation and analysis, narrowing the known bounds and suggesting potential for further improvement.

Contribution

It presents a novel band crossover heuristic for TSP tour construction, improving the upper bound estimate of the Euclidean TSP constant through simulation and concentration analysis.

Findings

Upper bound improved to approximately 0.90367

Simulation suggests future bounds limited to around 0.88

Heuristic could potentially lower the bound to about 0.85

Abstract

Consider $n$ points generated uniformly at random in the unit square, and let $L_n$ be the length of their optimal traveling salesman tour. Beardwood, Halton, and Hammersley (1959) showed $L_n / \sqrt n \to \beta$ almost surely as $n\to \infty$ for some constant $\beta$. The exact value of $\beta$ is unknown but estimated to be approximately $0.71$ (Applegate, Bixby, Chv\'atal, Cook 2011). Beardwood et al. further showed that $0.625 \leq \beta \leq 0.92116.$ Currently, the best known bounds are $0.6277 \leq \beta \leq 0.90380$, due to Gaudio and Jaillet (2019) and Carlsson and Yu (2023), respectively. The upper bound was derived using a computer-aided approach that is amenable to lower bounds with improved computation speed. In this paper, we show via simulation and concentration analysis that future improvement of the $0.90380$ is limited to $\sim0.88$. Moreover, we provide an alternative tour-constructing heuristic that, via simulation, could potentially improve the upper bound to $\sim0.85$. Our approach builds on a prior \emph{band-traversal} strategy, initially proposed by Beardwood et al. (1959) and subsequently refined by Carlsson and Yu (2023): divide the unit square into bands of height $\Theta(1/\sqrt{n})$, construct paths within each band, and then connect the paths to create a TSP tour. Our approach allows paths to cross bands, and takes advantage of pairs of points in adjacent bands which are close to each other. A rigorous numerical analysis improves the upper bound to $0.90367$.