论文标题
$ k $ - 换一种守望者路线
$k$-Transmitter Watchman Routes
论文作者
论文摘要
我们考虑了$ k $ - 递送器守望者的守望者路线问题：站在多边形$ p $中的点$ p $，如果$ \ overline {pq} $互相$ p $ p $ p $ t $ k $ times-$ q $是$ q $是$ k $ vissible，$ q $ wessible $ k $ to $ wessible $ k $ vissible to $ k $ co $ vissible of $ p $，watchman可以在P $中看到$ q \。沿着$ k $ - 租车守望者路线旅行，$ p $中的所有点或一组离散的点$ s \ subset p $必须为$ k $ - 守望者可视。我们的目的是最大程度地减少$ k $ - 租车守望者路线的长度。 我们表明，即使在简单的多边形中，最短的$ k $ - 传播者守望者路线问题对于离散点$ s \ subset p $ np-complete来说，也无法在对数因子内（除非p = np）近似，否则有或没有给定的起点。此外，我们提出了给定起点的$ k $ - 发射机守望员路由问题的聚类近似值和$ s \ subset p $，近似值$ o（\ log^2（| s | \ cdot n）\ log \ log \ log \ log \ log \ log \ log \ log \ log（| s | \ cdot n）\ cdot n）\ log（s | s | +1 |++++y $ | |
We consider the watchman route problem for a $k$-transmitter watchman: standing at point $p$ in a polygon $P$, the watchman can see $q\in P$ if $\overline{pq}$ intersects $P$'s boundary at most $k$ times -- $q$ is $k$-visible to $p$. Traveling along the $k$-transmitter watchman route, either all points in $P$ or a discrete set of points $S\subset P$ must be $k$-visible to the watchman. We aim for minimizing the length of the $k$-transmitter watchman route. We show that even in simple polygons the shortest $k$-transmitter watchman route problem for a discrete set of points $S\subset P$ is NP-complete and cannot be approximated to within a logarithmic factor (unless P=NP), both with and without a given starting point. Moreover, we present a polylogarithmic approximation for the $k$-transmitter watchman route problem for a given starting point and $S\subset P$ with approximation ratio $O(\log^2(|S|\cdot n) \log\log (|S|\cdot n) \log(|S|+1))$ (with $|P|=n$).
