Efficient Algorithms for Visibility-based Search in Polygonal Domains

Alapfogalmak

This paper studies bicriteria optimization problems for a single mobile agent within a polygonal domain, with the criteria of route length and area seen. It provides hardness results and approximation algorithms for the Quota Watchman Route Problem and the Budgeted Watchman Route Problem.

Kivonat

The paper investigates two visibility-based search problems in polygonal domains:

The Quota Watchman Route Problem (QWRP): Given a polygonal domain P and an area quota A, find a minimum length route that sees at least area A within P.
The Budgeted Watchman Route Problem (BWRP): Given a polygonal domain P and a length budget B, find a route that sees the maximum area within the budget constraint.

The key insights and results are:

The QWRP and BWRP are shown to be weakly NP-hard, even in simple polygons.
For the QWRP in a simple polygon, the paper provides the first fully polynomial-time approximation scheme (FPTAS) and a dual-approximation algorithm.
For the BWRP in a simple polygon, a polynomial-time approximation algorithm is given.
In polygonal domains with holes, hardness of approximation results and dual-approximation algorithms are provided.
The special case of a domain that is a union of lines is solved exactly in polynomial time for both problems.
The results also yield the first approximation algorithms for computing time-optimal search routes to guarantee a specified probability of detecting a static target randomly distributed within the domain.

Statisztikák

None.

Idézetek

None.

Főbb Kivonatok

Optimizing Visibility-based Search in Polygonal Domains

by Kien C. Huyn... : arxiv.org 04-03-2024

https://arxiv.org/pdf/2402.05420.pdf

Mélyebb kérdések

How can the algorithms be extended to handle multiple mobile agents searching the polygonal domain

To extend the algorithms for handling multiple mobile agents searching the polygonal domain, we can modify the objective function to optimize the routes for all agents collectively. This can involve finding a set of routes that collectively cover the maximum area while minimizing the total length traveled by all agents. The algorithm would need to consider the interactions between the routes of different agents to ensure efficient coverage of the polygonal domain. By introducing constraints and optimization criteria for multiple agents, the algorithm can be adapted to find optimal solutions for scenarios involving multiple mobile agents searching the domain simultaneously.

What are the implications of the hardness results on the practical applicability of these visibility-based search problems

The hardness results presented in the paper have significant implications for the practical applicability of visibility-based search problems. The NP-hardness of the Quota Watchman Route Problem (QWRP) and the Budgeted Watchman Route Problem (BWRP) in simple polygons indicate that finding exact solutions to these problems may be computationally challenging, especially as the size of the polygonal domain increases. This complexity suggests that approximation algorithms or heuristic approaches may be more suitable for real-world applications where efficiency is crucial. Additionally, the hardness results highlight the need for further research to develop efficient algorithms that can handle these optimization problems in a timely manner.

How can the techniques developed in this paper be applied to other geometric optimization problems involving visibility and coverage

The techniques developed in the paper for visibility-based search problems can be applied to other geometric optimization problems that involve visibility and coverage constraints. For example, in surveillance or patrolling scenarios, where mobile agents need to cover a certain area while minimizing the total distance traveled, the algorithms can be adapted to find optimal patrol routes. Similarly, in facility location or sensor placement problems, where the goal is to maximize coverage within a region, the concepts of visibility-based search can be utilized to optimize the placement of facilities or sensors. By leveraging the principles of visibility and coverage optimization, these techniques can be extended to a wide range of geometric optimization problems in various domains.