Learning-Augmented Algorithms for Online TSP on the Line

TL;DR

This paper develops learning-augmented algorithms for the online TSP on the line, achieving near-optimal competitive ratios with perfect predictions, robustness to errors, and graceful performance degradation.

Contribution

It introduces algorithms that are both competitive and robust, improving existing bounds for online TSP on the line with machine-learned predictions.

Findings

Achieves a 1.5 competitive ratio for the closed variant with perfect predictions.

Attains a 1.66 competitive ratio for the open variant with perfect predictions.

Provides bounds demonstrating robustness and smooth degradation as prediction errors increase.

Abstract

We study the online Traveling Salesman Problem (TSP) on the line augmented with machine-learned predictions. In the classical problem, there is a stream of requests released over time along the real line. The goal is to minimize the makespan of the algorithm. We distinguish between the open variant and the closed one, in which we additionally require the algorithm to return to the origin after serving all requests. The state of the art is a $1.64$-competitive algorithm and a $2.04$-competitive algorithm for the closed and open variants, respectively \cite{Bjelde:1.64}. In both cases, a tight lower bound is known \cite{Ausiello:1.75, Bjelde:1.64}. In both variants, our primary prediction model involves predicted positions of the requests. We introduce algorithms that (i) obtain a tight 1.5 competitive ratio for the closed variant and a 1.66 competitive ratio for the open variant in the case of perfect predictions, (ii) are robust against unbounded prediction error, and (iii) are smooth, i.e., their performance degrades gracefully as the prediction error increases. Moreover, we further investigate the learning-augmented setting in the open variant by additionally considering a prediction for the last request served by the optimal offline algorithm. Our algorithm for this enhanced setting obtains a 1.33 competitive ratio with perfect predictions while also being smooth and robust, beating the lower bound of 1.44 we show for our original prediction setting for the open variant. Also, we provide a lower bound of 1.25 for this enhanced setting.