Segment Watchman Routes

2026-06-24

AI Summary

This study addresses the multi-guard patrolling path problem by introducing a novel “segment guard path” model, which requires that every point inside a polygon be simultaneously visible from two distinct patrol paths, with the line segment connecting the two observing points fully contained within the polygon. The work formally defines this model for the first time and proves its NP-hardness under both min-max and min-sum optimization criteria. For simple polygons, the authors devise polynomial-time 2-approximation algorithms under each criterion and establish sufficient conditions for path validity. Integrating computational geometry, visibility analysis, and approximation algorithm design, this research provides a general framework for solving k-guard patrolling scenarios.

Abstract

Motivated by applications for robust guarding, we consider a variant of the multiple-watchmen problem that ensures that every point within a polygon $P$ is seen from more than one direction: we search for two routes $W_1,W_2$, such that every point $p\in P$ is contained in a segment $\overline{w_1w_2}\subseteq P$ such that $w_1\in W_1$ and $w_2\in W_2$. We call such routes segment watchman routes. We show that finding the two routes that are optimal with respect to the min-max criterion is weakly NP-hard even in simple polygons, and that finding the routes that are optimal with respect to the min-sum criterion is NP-hard in polygons with holes. Moreover, we present sufficient conditions for routes to be segment watchman routes, and provide a polynomial-time $2$-approximation under both the min-max criterion and the min-sum criterion, both in simple polygons. Finally, we show how to generalize our results for $k$ watchmen.

Problem

• segment watchman routes
• multiple-watchmen problem
• polygon guarding
• visibility coverage
• computational geometry

Innovation

• segment watchman routes
• multi-directional coverage
• approximation algorithm
• computational geometry
• NP-hardness