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 \(O(\log ^2(|S|\cdot n) \log \log (|S|\cdot n) \log |S|)\) (with \(|P|=n\)).
Supported by grants 2018-04001 (Nya paradigmer för autonom obemannad flygledning) and 2021-03810 (Illuminate: bevisbart goda algoritmer för bevakningsproblem) from the Swedish Research Council (Vetenskapsrådet).
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Nilsson, B.J., Schmidt, C. (2023). k-Transmitter Watchman Routes. In: Lin, CC., Lin, B.M.T., Liotta, G. (eds) WALCOM: Algorithms and Computation. WALCOM 2023. Lecture Notes in Computer Science, vol 13973. Springer, Cham. https://doi.org/10.1007/978-3-031-27051-2_18
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