Core Concepts
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.
Abstract
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.