Given an undirected graph, $G$, and vertices, $s$ and $t$ in $G$, the tracking paths problem is that of finding the smallest subset of vertices in $G$ whose intersection with any $s$-$t$ path results in a unique sequence. This problem is known to be NP-complete and has applications to animal migration tracking and detecting marathon course-cutting, but its approximability is largely unknown. In this paper, we address this latter issue, giving novel algorithms having approximation ratios of $(1+\epsilon)$, $O(\lg OPT)$ and $O(\lg n)$, for $H$-minor-free, general, and weighted graphs, respectively. We also give a linear kernel for $H$-minor-free graphs and make improvements to the quadratic kernel for general graphs.

翻译：考虑到一个未定向的图表,即$G$和odstate,$和$t$,追踪路径问题在于找到最小的1G$的脊椎子集,其与任何美元-美元路径的交叉以独特的顺序产生。这个问题已知是NP不完整的,适用于动物迁徙追踪和探测马拉松的分流切割,但其近似性却大为未知。在本文中,我们讨论后一问题,为普通图表提供近似率为$(1 ⁇ epsilon)$、O(lg Omplo)$和O(lgn)$(lg n)的新型算法,用于以H$-minor-free、一般和加权的图表。我们还为1H$-minor-crole 图形提供直线内核,并改进通用图形的四角内核。