The following geometric vehicle scheduling problem has been considered: given continuous curves $f_1, \ldots, f_n : \mathbb{R} \rightarrow \mathbb{R}^2$, find non-negative delays $t_1, \ldots, t_n$ minimizing $\max \{ t_1, \ldots, t_n \}$ such that, for every distinct $i$ {and $j$} and every time $t$, $| f_j (t - t_j) - f_i (t - t_i) | > \ell$, where~$\ell$ is a given safety distance. We study a variant of this problem where we consider trains (rods) of fixed length $\ell$ that move at constant speed and sets of train lines (tracks), each of which consisting of an axis-parallel line-segment with endpoints in the integer lattice $\mathbb{Z}^d$ and of a direction of movement (towards $\infty$ {or $- \infty$}). We are interested in upper bounds on the maximum delay we need to introduce on any line to avoid collisions, but more specifically on universal upper bounds that apply no matter the set of train lines. We show small universal constant upper bounds for $d = 2$ and any given $\ell$ and also for $d = 3$ and $\ell = 1$. Through clique searching, we are also able to show that several of these upper bounds are tight.

中文翻译：

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恒定延迟格子列车时刻表

考虑了以下几何车辆调度问题：给定连续曲线 $f_1, \ldots, f_n : \mathbb{R} \rightarrow \mathbb{R}^2$，找到非负延迟 $t_1, \ldots, t_n$最小化 $\max \{ t_1, \ldots, t_n \}$ 这样，对于每个不同的 $i$ {and $j$} 和每次 $t$, $| f_j (t - t_j) - f_i (t - t_i) | > \ell$，其中~$\ell$ 是给定的安全距离。我们研究了这个问题的一个变体，我们考虑以恒定速度移动的固定长度 $\ell$ 的火车（杆）和多组火车线（轨道），每条线都由一个轴平行线段组成，端点为整数格 $\mathbb{Z}^d$ 和运动方向（朝向 $\infty$ {或 $- \infty$}）。我们对需要在任何线路上引入以避免冲突的最大延迟的上限感兴趣，但更具体地说，适用于无论火车线路集如何的通用上限。我们展示了 $d = 2$ 和任何给定的 $\ell$ 以及 $d = 3$ 和 $\ell = 1$ 的小通用常数上限。通过集团搜索，我们还能够证明这些上限中有几个是紧的。