Combinatorics and complexity in geometric visibility problems

Abstract

Geometric visibility is fundamental to computational geometry and its applications in areas such as robotics, sensor networks, CAD, and motion planning. We explore combinatorial and computational complexity problems arising in a collection of settings that depend on various notions of visibility. We first consider a generalized version of the classical art gallery problem in which the input specifies the number of reflex vertices r and convex vertices c of the simple polygon ( n = r + c). This additional information better characterizes the shape of the polygon. Through a lower bound construction, tight combinatorial bounds for coverage are achieved for all r ≥ 0 and c ≥ 3. The combinatorics of guarding polyominoes and other polyforms are studied in terms of m, the number of cells, as opposed to the traditional parameter n. Various visibility models and guard types are considered. We establish that finding a minimum cardinality guard set for covering a polyomino is NP-hard. We introduce an algorithm for constructing a spiral serpentine polygonization of a set of n ≥ 3 points in the plane. The algorithm's behavior can be viewed as incrementally appending a visible triangle to the triangulation constructed so far. We consider beacon-based point-to-point routing and coverage problems. A beacon b is a point that can be activated to effect a gravitational pull toward itself in a polygonal domain. Algorithms are given for computing the attraction region of b and finding a minimum size set of beacons to route from a source s to a destination t given a finite set of candidate beacon locations. We show that finding a minimum cardinality set of beacons to cover a simple polygon or conduct certain types of routing in a simple polygon is NP-hard.