$k$-发射器守望者路线
$k$-Transmitter Watchman Routes
AI总结：
研究$k$-发射器守望者路线问题，证明在简单多边形中求最短路线为NP完全且难以近似，并给出给定起点时的多对数近似算法。
摘要：
我们考虑$k$-发射器守望者的守望者路线问题：站在多边形$P$中的点$p$处，守望者可以看到$q\in P$，如果线段$\overline{pq}$与$P$的边界相交至多$k$次——即$q$对$p$是$k$-可见的。沿着$k$-发射器守望者路线行进时，$P$中的所有点或离散点集$S\subset P$必须对守望者$k$-可见。我们的目标是使$k$-发射器守望者路线的长度最小化。我们证明，即使在简单多边形中，对于离散点集$S\subset P$的最短$k$-发射器守望者路线问题是NP完全的，并且（除非P=NP）无法在对数因子内近似，无论是否给定起点。此外，我们针对给定起点和$S\subset P$的$k$-发射器守望者路线问题提出了一个多对数近似，近似比为$O(\log^2(|S|\cdot n) \log\log (|S|\cdot n) \log(|S|+1))$（其中$|P|=n$）。
英文摘要：
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 $|P|=n$).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
