  We initiate the study of online routing problems with predictions, inspired
by recent exciting results in the area of learning-augmented algorithms. A
learning-augmented online algorithm which incorporates predictions in a
black-box manner to outperform existing algorithms if the predictions are
accurate while otherwise maintaining theoretical guarantees even when the
predictions are extremely erroneous is a popular framework for overcoming
pessimistic worst-case competitive analysis.
  In this study, we particularly begin investigating the classical online
traveling salesman problem (OLTSP), where future requests are augmented with
predictions. Unlike the prediction models in other previous studies, each
actual request in the OLTSP, associated with its arrival time and position, may
not coincide with the predicted ones, which, as imagined, leads to a
troublesome situation. Our main result is to study different prediction models
and design algorithms to improve the best-known results in the different
settings. Moreover, we generalize the proposed results to the online
dial-a-ride problem.
