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

TL;DR

This work tackles the Euclidean TSP constant $\beta$ for $n$ random points in the unit square by building on the band-traversal framework and incorporating band-crossing opportunities. It couples Monte Carlo simulations with concentration inequalities to quantify finite-sample deviations in the tuple-optimization approach, achieving a rigorous bound $\beta\le0.90367$ and demonstrating a band-crossing heuristic that empirically lowers the upper bound toward $0.85$ for larger tuple sizes. The authors also show that purely tuple-optimization strategies likely plateau around $0.86$–$0.88$, suggesting diminishing returns with larger $k$ given computational constraints. Overall, the paper provides both a small but robust numerical improvement and a principled, certifiable enhancement method that points to further avenues beyond band-constrained traversals, with potential practical impact on understanding the asymptotic constant in probabilistic TSP settings.

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 $L_n / \sqrt n \to β$ almost surely as $n\to \infty$ for some constant $β$. The exact value of $β$ is unknown but estimated to be approximately $0.71$ (Applegate, Bixby, Chvátal, Cook 2011). Beardwood et al. further showed that $0.625 \leq β\leq 0.92116.$ Currently, the best known bounds are $0.6277 \leq β\leq 0.90380$, 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 $Θ(1/\sqrt{n})$, 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 $0.90367$.

Figures (3)

• Figure 1: By allowing band crossovers, there exist $3$ options to traverse the vertices in a $(k+1)$-tuple when $k=3$. The brown path is induced by the identity permutation $1234$, the blue path is induced by the permutation $1324$, and the pink path traverses the first, third, and fourth vertices in the tuple while appending vertex $u$ to the adjacent band.
• Figure 2: An example configuration satisfying $\mathcal{S}$ for $k=4$
• Figure 3: Possible configurations of $u, v, w$.

Theorems & Definitions (9)

• Theorem 1.1: BHH Theorem
• Lemma 2.1: BHH
• Theorem 3.1
• proof : Proof of Theorem \ref{['thm:conc_beta_chernoff_correct']}
• Definition 3.2
• Lemma 3.3: concentration_inequalities, Section 2.4
• Corollary 4.1
• Lemma 4.1
• proof