We study a new version of the Euclidean TSP called VectorTSP (VTSP for short) where a mobile entity is allowed to move according to a set of physical constraints inspired from the pen-and-pencil game Racetrack (also known as Vector Racer ). In contrast to other versions of TSP accounting for physical constraints, such as Dubins TSP, the spirit of this model is that (1) no speed limitations apply, and (2) inertia depends on the current velocity. As such, this model is closer to typical models considered in path planning problems, although applied here to the visit of n cities in a non-predetermined order. We motivate and introduce the VectorTSP problem, discussing fundamental differences with previous versions of TSP. In particular, an optimal visit order for ETSP may not be optimal for VTSP. We show that VectorTSP is NP-hard, and in the other direction, that VectorTSP reduces to GroupTSP in polynomial time (although with a significant blow-up in size). On the algorithmic side, we formulate the search for a solution as an interactive scheme between a high-level algorithm and a trajectory oracle, the former being responsible for computing the visit order and the latter for computing the cost (or the trajectory) for a given visit order. We present algorithms for both, and we demonstrate and quantify through experiments that this approach frequently finds a better solution than the optimal trajectory realizing an optimal ETSP tour, which legitimates the problem itself and (we hope) motivates further algorithmic developments.

中文翻译：

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VectorTSP：具有类似赛道加速约束的旅行销售员问题

我们研究了一个新版本的欧几里得 TSP，称为 VectorTSP（简称为 VTSP），其中允许移动实体根据一组受钢笔游戏 Racetrack（也称为 Vector Racer）启发的物理约束进行移动。与考虑物理约束的其他 TSP 版本（例如 Dubins TSP）相比，该模型的精神是 (1) 不适用速度限制，以及 (2) 惯性取决于当前速度。因此，该模型更接近路径规划问题中考虑的典型模型，尽管这里应用于以非预定顺序访问 n 个城市。我们激发并介绍了 VectorTSP 问题，讨论了与以前版本的 TSP 的根本区别。特别是，ETSP 的最佳访问顺序对于 VTSP 可能不是最佳的。我们证明 VectorTSP 是 NP 难的，在另一个方向上，VectorTSP 在多项式时间内减少到 GroupTSP（尽管在大小上有显着的膨胀）。在算法方面，我们将寻找解决方案制定为高级算法和轨迹预言机之间的交互方案，前者负责计算访问顺序，后者负责计算访问次数（或轨迹）。给定的访问顺序。我们为两者都提供了算法，并且我们通过实验证明和量化了这种方法经常找到比实现最佳 ETSP 旅行的最佳轨迹更好的解决方案，这使问题本身合法化并且（我们希望）激发进一步的算法发展。我们将搜索解决方案制定为高级算法和轨迹预言机之间的交互方案，前者负责计算访问顺序，后者负责计算给定访问顺序的成本（或轨迹）。我们为两者都提供了算法，并且我们通过实验证明和量化了这种方法经常找到比实现最佳 ETSP 旅行的最佳轨迹更好的解决方案，这使问题本身合法化并且（我们希望）激发进一步的算法发展。我们将搜索解决方案制定为高级算法和轨迹预言机之间的交互方案，前者负责计算访问顺序，后者负责计算给定访问顺序的成本（或轨迹）。我们为两者都提供了算法，并且我们通过实验证明和量化了这种方法经常找到比实现最佳 ETSP 旅行的最佳轨迹更好的解决方案，这使问题本身合法化并且（我们希望）激发进一步的算法发展。