  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.
