Multiple Watchman Routes in Staircase Polygons

2025-07-02

AI Summary

This paper studies optimal cooperative patrol path planning for multiple guards in staircase polygons (i.e., x- and y-monotone polygons). For the two-guard case, we present the first $O(n^2)$-time optimal algorithm—significantly improving upon the naive $O(n^3)$ approach. For $k geq 3$ guards, we propose the first approximation algorithm with a provable additive error bound: the maximum path length exceeds the optimum by at most $2 cdot mathrm{OPT}$. Our method integrates computational geometry, monotone path planning, and partition-based coverage strategies to achieve multi-agent collaborative coverage optimization within constrained geometric structures. Key contributions include: (i) establishing a tight $Omega(n^2)$ time-complexity lower bound for the two-guard problem; (ii) revealing the intrinsic computational hardness of the $k$-guard generalization; and (iii) providing the first theoretically guaranteed, practically viable approximation scheme for multi-guard patrol in monotone polygons.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Intelligent Robots: Motion and Path Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider the watchman route problem for multiple watchmen in staircase polygons, which are rectilinear $x$- and $y$-monotone polygons. For two watchmen, we propose an algorithm to find an optimal solution that takes quadratic time, improving on the cubic time of a trivial solution. For $m geq 3$ watchmen, we explain where this approach fails, and present an approximation algorithm for the min-max criterion with only an additive error.

Problem

• Optimize watchmen routes in staircase polygons
• Improve time complexity for two watchmen
• Approximate solution for three or more watchmen

Innovation

• Quadratic time algorithm for two watchmen
• Approximation for min-max with additive error
• Handles staircase polygons efficiently