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.

翻译：以下几何车辆调度问题已得到考虑:考虑到连续的曲线$f_1,\ldots, f_n:\mathbb{R}\rightrow\mathbb{R}\rrightrow\mathb{R#2$,发现非负延迟$t_1,\ldots, t_n$最小化美元=maxxx美元=t_1,\ldots, t_n}美元,因此,对于每一个不同的美元 {和美元}和每时美元, 美元 f_j (t -t_j) -f_i (t -t_i)\right_rormaxxxy)\\\\\\ lixxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx