Contiguous Boundary Guarding

2024-12-19

arXiv.org

AI Summary

This paper studies the *continuous segment guarding problem* for simple polygons: minimizing the number of guards required to cover the entire polygon boundary, under the constraint that each guard can cover only a single contiguous arc of the boundary. Contrary to the classical (unconstrained) art gallery problem—which is NP-hard—we present the first polynomial-time exact algorithm for this continuity-constrained variant. We rigorously establish a tight bound of ⌊(n−2)/2⌋ on the optimal number of guards, and construct explicit instances achieving this bound. Additionally, we propose a greedy approximation algorithm with additive error at most one (i.e., outputting at most OPT + 1 guards). Our key insight is that the continuity constraint—often perceived as restrictive—actually simplifies the problem structurally. We introduce a novel combinatorial characterization of feasible guard placements and develop an efficient algorithmic framework grounded in this structure.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $lfloor frac{n-2}{2} floor$ guards. This bound is tight because there are polygons that require this many guards.

Problem

• Minimize guards for contiguous polygon boundary coverage
• Develop exact polynomial-time algorithm for optimal guarding
• Establish tight upper bound on guards for n-vertex polygons

Innovation

• Greedy algorithm for contiguous boundary guarding
• Polynomial-time exact algorithm solution
• Tight upper bound on guard count