Optimizing Visibility-based Search in Polygonal Domains
CoRR(2024)
摘要
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.
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