Contiguous Boundary Guarding

TL;DR

The paper addresses contiguous boundary guarding of a simple polygon, a variant of the art gallery problem where each guard covers a contiguous boundary portion. It introduces a greedy strategy with a bound of $OPT+1$ and an exact polynomial-time algorithm achieved by constructing a polynomial-size starting set $S$ based on reflex-vertex-driven candidates, guaranteeing optimality when evaluated from all $p\in S$. A tight combinatorial bound of $\left\lfloor \frac{n-2}{2} \right\rfloor$ guards is proved and shown to be achievable for all $n$, underscoring the problem's tractability under contiguity constraints. Overall, the work advances both combinatorial understanding and algorithmic techniques for a natural, constrained variant of a classical hard problem in computational geometry.

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.

Figures (14)

• Figure 1: Illustration of the proof of Lemma \ref{['cover-lemma']}.
• Figure 2: Illustration of the lower bound $\lfloor\frac{n-2}{2}\rfloor$ guards. The polygonal arcs around the boundary are maximal chains that can each be guarded by a single guard.
• Figure 3: The boundary of the polygon is assumed to be directed counter-clockwise.
• Figure 4: Illustration of the proof of Theorem \ref{['greedy-thr']}. (a) $p_1$ is covered by one guard $g^*_1$ in an optimal solution. (b) $p_1$ is covered by two guards $g^*_1$ and $g^*_2$ in an optimal solution.
• Figure 5: Illustration of the proof of Lemma \ref{['on-ray-lemma']}.

Theorems & Definitions (29)

• Lemma 1
• proof
• Theorem 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof