Vector TSP: A Traveling Salesperson Problem with Racetrack-like Acceleration Constraints

Abstract: We study a new version of the Traveling Salesperson Problem, called \VectorTSP, where the traveler is subject to discrete acceleration constraints, as defined in the paper-and-pencil game Racetrack (also known as Vector Racer). In this model, the degrees of freedom at a certain point in time depends on the current velocity, and the speed is not limited. The paper introduces this problem and initiates its study, discussing also the main differences with existing versions of TSP. Not surprisingly, the problem turns out to be NP-hard. A key feature of \VectorTSP is that it deals with acceleration in a discrete, combinatorial way, making the problem more amenable to algorithmic investigation. The problem involves two layers of trajectory planning: (1) the order in which cities are visited, and (2) the physical trajectory realizing such a visit, both interacting with each other. This interaction is formalized as an interactive protocol between a high-level tour algorithm and a trajectory oracle, the former calling the latter repeatedly. We present an exact implementation of the trajectory oracle, adapting the A* algorithm for paths over multiple checkpoints whose ordering is \emph{given} (this algorithm being possibly of independent interest). To motivate the problem further, we perform experiments showing that the naive approach consisting of solving the instance as an \EuclideanTSP first, then optimizing the trajectory of the resulting tour, is typically suboptimal and outperformed by simple (but dedicated) heuristics.