  We consider the time-dependent traveling salesman problem (TDTSP), a generalization of the asymmetric traveling salesman problem (ATSP) to incorporate time-dependent cost functions. In our model, the costs of an arc can change arbitrarily over time (and do not only dependent on the position in the tour). The TDTSP turns out to be structurally more difficult than the TSP. We prove it is NP-hard and APX-hard even if a generalized version of the triangle inequality is satisfied. In particular, we show that even the computation of one-trees becomes intractable in the case of time-dependent costs. We derive two IP formulations of the TDTSP based on time-expansion and propose different pricing algorithms to handle the significantly in- creased problem size. We introduce multiple families of cutting planes for the TDTSP as well as different LP-based primal heuristics, a propaga- tion method and a branching rule. We conduct computational experiments to evaluate the effectiveness of our approaches on randomly generated in- stances. We are able to decrease the optimality gap remaining after one hour of computations to about six percent, compared to a gap of more than forty percent obtained by an off-the-shelf IP solver. Finally, we carry out a first attempt to learn strong branching decisions for the TDTSP. At the current state, this method does not improve the running times. 
Cuts, Primal Heuristics, and Learning to Branch for the Time-Dependent   Traveling Salesman Problem