$k$-Transmitter Watchman Routes

TL;DR

This paper studies the $k$-transmitter watchman route problem, proving its computational hardness and providing a polylogarithmic approximation algorithm for route minimization in polygons.

Contribution

It establishes NP-completeness and inapproximability results, and introduces a polylogarithmic approximation algorithm for the $k$-transmitter watchman route problem.

Findings

Shortest route problem is NP-complete for discrete points.

Cannot be approximated within a logarithmic factor unless P=NP.

Provides a polylogarithmic approximation algorithm with ratio $O( ext{log}^2(|S|n) ext{log} ext{log}(|S|n) ext{log}(|S|+1))$.

Abstract

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 $|P|=n$).