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

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 $0.90367$.

💡 Research Summary

The paper addresses the long‑standing problem of tightening the upper bound for the Euclidean traveling‑salesman constant β, defined by the Beardwood‑Halton‑Hammersley theorem as the limit of Lₙ/√n for n uniformly random points in the unit square. The current best bounds are 0.6277 ≤ β ≤ 0.90380, the latter obtained by Carlsson and Yu (2024) through a “tuple‑optimization” scheme. That method partitions the square into horizontal bands of height h/√n, groups points within each band into (k + 1)‑tuples, and selects the permutation of each tuple that minimizes the intra‑band path length while keeping the leftmost and rightmost points fixed. By increasing k, the bound can be lowered, but the computational cost grows exponentially.

The authors first evaluate how far this approach can be pushed. Using massive Monte‑Carlo simulations (up to 10⁷ replicates) and a concentration analysis based on sub‑gamma and Hoeffding inequalities, they estimate that even with k = 8 the tuple method yields an asymptotic bound around 0.862, and that the practical limit of the method is roughly 0.88. Thus, further improvements via pure tuple optimization alone appear marginal.

To break this barrier, the paper introduces a new “band‑crossing” heuristic. The key idea is to allow vertices that lie close to a band boundary to be paired with vertices in the adjacent band, forming a short 2‑cycle across the boundary. Although a 2‑cycle does not constitute a valid TSP tour, the authors show that it can be transformed into a feasible tour by removing one of its edges and reconnecting the remaining vertices, with the triangle inequality guaranteeing that the total length does not increase. For each (k + 1)‑tuple the algorithm compares three options: the original left‑to‑right order, the optimal permutation found by tuple optimization, and the crossing option. The crossing is chosen whenever it yields a shorter path.

Simulation results demonstrate that this strategy consistently improves the bound: for k = 4 the estimated upper bound drops to ≈ 0.869, and for k = 8 to ≈ 0.850, both noticeably better than the best tuple‑only results. Moreover, by performing a rigorous numerical integration using a decision‑tree partition of the integration domain (as in Carlsson and Yu) but now incorporating the crossing option, the authors obtain a provably valid upper bound β ≤ 0.90367, improving the previous record by 0.00013.

The paper concludes that the tuple‑optimization framework is essentially capped near 0.88, and that substantial further reductions of β will require structural innovations such as band crossings. It also highlights that the proposed heuristic is compatible with any future improvements in the tuple method, so advances in computational power or algorithmic design could be immediately leveraged to push the bound toward the conjectured value around 0.71.