Online Metric TSP

Abstract

In the online metric traveling salesperson problem, n points of a metric space arrive one by one and have to be placed (immediately and irrevocably) into empty cells of a size-n array. The goal is to minimize the sum of distances between consecutive points in the array. This problem was introduced by Abrahamsen, Bercea, Beretta, Klausen, and Kozma [ESA'24] as a generalization of the online sorting problem, which was introduced by Aamand, Abrahamsen, Beretta, and Kleist [SODA'23] as a tool in their study of online geometric packing problems.
Online metric TSP has been studied for a range of fixed metric spaces. For 1-dimensional Euclidean space, the problem is equivalent to online sorting, where an optimal competitive ratio of Θ(√n) is known. For d-dimensional Euclidean space, the best-known upper bound is O(2^d √{dn log n}), leaving a gap to the Ω(√n) lower bound. Finally, for the uniform metric, where all distances are 0 or 1, the optimal competitive ratio is known to be Θ(log n).
We study the problem for a general metric space, presenting an algorithm with competitive ratio O(√n). In particular, we close the gap for d-dimensional Euclidean space, completely removing the dependence on dimension. One might hope to simultaneously guarantee competitive ratio O(√n) in general and O(log n) for the uniform metric, but we show that this is impossible.