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arXiv · 2412.15053

Contiguous Boundary Guarding

Ahmad Biniaz·Anil Maheshwari·Joseph S. B. Mitchell·Saeed Odak·Valentin Polishchuk·Thomas Shermer
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Abstract

We study the problem of guarding the boundary of a simple polygon with a minimum number of guards such that each guard covers a contiguous portion of the boundary. First, we present a simple greedy algorithm for this problem that returns a guard set of size at most OPT + 1, where OPT is the number of guards in an optimal solution. Then, we present a polynomial-time exact algorithm. While the algorithm is not complicated, its correctness proof is rather involved. This result is interesting in the sense that guarding problems are typically NP-hard and, in particular, it is NP-hard to minimize the number of guards to see the boundary of a simple polygon, without the contiguous boundary guarding constraint. From the combinatorial point of view, we show that any $n$-vertex polygon can be guarded by at most $\lfloor \frac{n-2}{2}\rfloor$ guards. This bound is tight because there are polygons that require this many guards.

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Ahmad Biniaz, Anil Maheshwari, Joseph S. B. Mitchell, Saeed Odak, Valentin Polishchuk, Thomas Shermer. 2024-12-19. Contiguous Boundary Guarding. https://arxiv.org/abs/2412.15053

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