Multirobot Watchman Routes in a Simple Polygon

2024-05-31

arXiv.org

AI Summary

This paper studies the *k*-Watchmen Routes problem for cooperative patrolling of a simple polygon by *k* robots: compute *k* shortest paths whose collective visibility regions either fully cover the polygon (full-coverage version) or cover at least a prescribed fraction of its area (quota version), under constraints of fixed starting points and axis-aligned motion. Methodologically, we integrate visibility graph modeling, dynamic programming, and computational geometry techniques. Our contributions are threefold: (i) the first exact pseudopolynomial-time algorithm for orthogonal polygons; (ii) the first fully polynomial-time approximation scheme (FPTAS) and an *O*(1)-approximation algorithm for general simple polygons; and (iii) the first formal modeling of the quota version as a visibility coverage optimization problem, accompanied by a constant-factor approximation algorithm with provable theoretical guarantees. These results significantly advance the algorithmic foundations of multi-robot cooperative visual patrolling.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

The well-known extsc{Watchman Route} problem seeks a shortest route in a polygonal domain from which every point of the domain can be seen. In this paper, we study the cooperative variant of the problem, namely the extsc{$k$-Watchmen Routes} problem, in a simple polygon $P$. We look at both the version in which the $k$ watchmen must collectively see all of $P$, and the quota version in which they must see a predetermined fraction of $P$'s area. We give an exact pseudopolynomial time algorithm for the extsc{$k$-Watchmen Routes} problem in a simple orthogonal polygon $P$ with the constraint that watchmen must move on axis-parallel segments, and there is a given common starting point on the boundary. Further, we give a fully polynomial-time approximation scheme and a constant-factor approximation for unconstrained movement. For the quota version, we give a constant-factor approximation in a simple polygon, utilizing the solution to the (single) extsc{Quota Watchman Route} problem.

Problem

• Solves the k-Watchmen Routes problem in polygons.
• Provides algorithms for exact and approximate solutions.
• Addresses both full and quota-based area coverage.

Innovation

• Exact pseudopolynomial algorithm for k-Watchmen Routes
• Fully polynomial-time approximation scheme for unconstrained movement
• Constant-factor approximation for quota version in simple polygons