Contiguous Boundary Guarding

TL;DR

This paper introduces efficient algorithms for guarding a polygon's boundary with minimal guards, ensuring each guard covers a contiguous boundary segment, and proves optimal guard bounds for polygons.

Contribution

It presents a simple greedy approximation and a polynomial-time exact algorithm for contiguous boundary guarding, a problem previously thought to be NP-hard.

Findings

Greedy algorithm achieves guard set size at most OPT + 1.

Polynomial-time exact algorithm with a complex correctness proof.

Any n-vertex polygon can be guarded with at most ⌊(n-2)/2⌋ guards.

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.