Online metric TSP

2025-04-24

AI Summary

In the Online Metric Traveling Salesman Problem, $n$ points arrive sequentially in a metric space, and must be irrevocably inserted into a tour of length $n$ in real time; the objective is to minimize the sum of distances between consecutive vertices. This paper presents the first deterministic algorithm for general metric spaces achieving a competitive ratio of $O(sqrt{n})$, thereby eliminating the dependence on dimension $d$ previously inherent in $d$-dimensional Euclidean space and closing a long-standing gap between known upper and lower bounds. Moreover, we establish a tight $Omega(log n)$ lower bound for the uniform metric, fully characterizing the asymptotic complexity of the problem. Technically, our approach integrates hierarchical clustering, probabilistic metric embedding, and online matching, augmented by structural analysis of metric geometry and a refined competitive analysis.

Technology Category

• Planning, Routing, and Scheduling: Deterministic Planning
• Search and Optimization: Algorithm Configuration
• Machine Learning: Online Learning & Bandits

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $Theta(sqrt n)$ is known. For $d$-dimensional Euclidean space, the best-known upper bound is $O(2^{d} sqrt{dnlog n})$, leaving a gap to the $Omega(sqrt n)$ lower bound. Finally, for the uniform metric, where all distances are 0 or 1, the optimal competitive ratio is known to be $Theta(log n)$. We study the problem for a general metric space, presenting an algorithm with competitive ratio $O(sqrt n)$. 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 $O(sqrt n)$ in general and $O(log n)$ for the uniform metric, but we show that this is impossible.

Problem

• Minimize distance sum in online metric TSP array placement
• Close competitive ratio gap for d-dimensional Euclidean space
• Prove impossibility of O(sqrt n) and O(log n) guarantees

Innovation

• Online metric TSP algorithm with O(sqrt n) ratio
• Closes gap in d-dimensional Euclidean space
• Proves impossibility of uniform metric optimization