Optimizing Visibility-based Search in Polygonal Domains

2024-02-08

Scandinavian Workshop on Algorithm Theory

AI Summary

This paper addresses the visibility-based search path optimization problem for a single agent in planar polygonal domains (with holes) and unions of line segments, optimizing simultaneously for path length and visible area. Three objectives are considered: (i) minimizing path length subject to a minimum visible area constraint; (ii) maximizing visible area under a path length budget; and (iii) computing a time-optimal search path under a probabilistic detection model with prior target distribution and prescribed detection probability. We present the first fully polynomial-time approximation scheme (FPTAS) for visible area quota-constrained search in simple polygons, along with a more efficient dual approximation algorithm. These results are extended to polygons with holes and to unions of line segments. We prove NP-hardness of the problem and provide rigorous polynomial-time approximation guarantees. Notably, this work delivers the first theoretically grounded, time-optimal search path computation under probabilistic detection—establishing both hardness and approximability for visibility search across complex planar domains.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Agentic search
• Security and Privacy: Large-scale security measurements

Abstract

Given a geometric domain $P$, visibility-based search problems seek routes for one or more mobile agents ("watchmen") to move within $P$ in order to be able to see a portion (or all) of $P$, while optimizing objectives, such as the length(s) of the route(s), the size (e.g., area or volume) of the portion seen, the probability of detecting a target distributed within $P$ according to a prior distribution, etc. The classic watchman route problem seeks a shortest route for an observer, with omnidirectional vision, to see all of $P$. In this paper we study bicriteria optimization problems for a single mobile agent within a polygonal domain $P$ in the plane, with the criteria of route length and area seen. Specifically, we address the problem of computing a minimum length route that sees at least a specified area of $P$ (minimum length, for a given area quota). We also study the problem of computing a length-constrained route that sees as much area as possible. We provide hardness results and approximation algorithms. In particular, for a simple polygon $P$ we provide the first fully polynomial-time approximation scheme for the problem of computing a shortest route seeing an area quota, as well as a (slightly more efficient) polynomial dual approximation. We also consider polygonal domains $P$ (with holes) and the special case of a planar domain consisting of a union of lines. Our results yield the first approximation algorithms for computing a time-optimal search route in $P$ to guarantee some specified probability of detection of a static target within $P$, randomly distributed in $P$ according to a given prior distribution.

Problem

• Optimize visibility-based search in polygonal domains.
• Compute minimum length routes for specified area coverage.
• Develop approximation algorithms for time-optimal search routes.

Innovation

• Develops approximation algorithms for visibility-based search.
• Optimizes route length and visible area in polygonal domains.
• Introduces polynomial-time schemes for area quota routes.