一种改进的跟踪路径问题的核函数
Improved Kernels for Tracking Path Problem
摘要：
移动物体的跟踪对于安全系统和网络至关重要。给定一个图$G$，终端顶点$s$和$t$，以及一个整数$k$，\textsc{Tracking Paths}问题询问是否存在最多$k$顶点，如果标记为跟踪器，将确保在每个s-t路径中遇到的跟踪器序列是唯一的。已知该问题是NP-难的，并且当通过输出（跟踪集）$k$的大小进行参数化时，允许具有$\mathcal{O}(k^6)$顶点和$\mathcal{O}(k^7)$边的核（可简化为等价实例）[5]。一个有趣的问题是，现有的内核是否可以改进。在本文中，我们肯定地回答这个问题：（i）对于一般的图，我们证明了存在一个核的大小为$\mathcal{O}(k^2)$，（ii）对于平面图，我们进一步改进这一点，给出一个核的大小为$\mathcal{O}(k)$。此外，我们还证明了当用$k$参数化时，对于参数化复杂性类W[1]，对于$n$顶点上的图，找到一个最大为$n-k$的跟踪集是困难的.
英文摘要：
Tracking of moving objects is crucial to security systems and networks. Given a graph $G$, terminal vertices $s$ and $t$, and an integer $k$, the \textsc{Tracking Paths} problem asks whether there exists at most $k$ vertices, which if marked as trackers, would ensure that the sequence of trackers encountered in each s-t path is unique. It is known that the problem is NP-hard and admits a kernel (reducible to an equivalent instance) with $\mathcal{O}(k^6)$ vertices and $\mathcal{O}(k^7)$ edges, when parameterized by the size of the output (tracking set) $k$ [5]. An interesting question that remains open is whether the existing kernel can be improved. In this paper we answer this affirmatively: (i) For general graphs, we show the existence of a kernel of size $\mathcal{O}(k^2)$, (ii) For planar graphs, we improve this further by giving a kernel of size $\mathcal{O}(k)$. In addition, we also show that finding a tracking set of size at most $n-k$ for a graph on $n$ vertices is hard for the parameterized complexity class W[1], when parameterized by $k$.
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