Stochastic Online Euclidean TSP

Abstract

In the Euclidean travelling salesman problem (Euclidean TSP), a salesman must visit $n$ points in Euclidean space, while minimizing the travel distance, according to the Euclidean distance function. In online Euclidean TSP, introduced by Abrahamsen, Bercea, Beretta, Klausen and Kozma [ESA 2024], the points are revealed one at a time, and a time slot must be assigned before the next is revealed. Once a point is assigned to a time slot, it can never be reassigned to another time slot. There are $n$ time slots. Euclidean online TSP is a high-dimensional generalization of online sorting, introduced by Aamand, Abrahamsen, Beretta and Kleist [SODA 2023]. Bertram [ESA 2025] showed an algorithm that achieves a competitive ratio of $O(\sqrt{n})$ in the worst case. In stochastic online Euclidean TSP, the points are sampled uniformly and independently in the unit $d$-cube. Kalavas, Platanos and Tolias [STACS 2026] presented an algorithm achieving a competitive ratio of $O(\log^2 n)$ with high probability for stochastic online Euclidean TSP. We present a simple algorithm that for $d \geq 2$ achieves an expected competitive ratio of $O(1)$, and for $d=1$ achieves an expected competitive ratio of $O(\log n)$, matching the lower bound by Hu [SODA 2026] for $d=1$. The algorithm is deterministic, and the expectation is due to the stochastic input. We also show that in the variant where there are more time slots than points, i.e., $\left\lceil(1 + \varepsilon)n\right\rceil$ time slots and $d=1$, our algorithm achieves an expected competitive ratio of $O\left(1 + \log \varepsilon^{-1}\right)$. We also survey algorithms from the literature. We experimentally evaluate our algorithm, which reveals that in all variants the constant factor hidden asymptotically is small. We also evaluate the algorithms from the literature.