Abstract
Given a set [Formula: see text] of [Formula: see text] points in the plane, how many universal guards are sometimes necessary and always sufficient to guard any simple polygon with vertex set [Formula: see text]? We call this problem a Universal Guard Problem and provide a spectrum of results. We give upper and lower bounds on the number of universal guards that are always sufficient to guard all polygons having a given set of [Formula: see text] vertices, or to guard all polygons in a given set of [Formula: see text] polygons on an [Formula: see text]-point vertex set. Our upper bound proofs include algorithms to construct universal guard sets of the respective cardinalities.
Highlights
Problems of finding optimal covers are among the most fundamental algorithmic challenges that play an important role in many contexts
While Klee’s question was posed about guarding an n-vertex simple polygon, a related question about point sets was posed at the 2014 NYU Goodman-Pollack Fest: Given a set S of n points in the plane, how many universal guards are sometimes necessary and always sufficient to guard any simple polygon with vertex set S? This problem, and several related
Our Universal Guard Problem is, in a sense, an extreme version of the problem of guarding a set of possible polygonalizations that are consistent with a given set of sample points that are the polygon vertices: In the universal setting, we require that the guards are a rich enough set to achieve visibility coverage for all possible polygonalizations
Summary
Problems of finding optimal covers are among the most fundamental algorithmic challenges that play an important role in many contexts. Our Universal Guard Problem is, in a sense, an extreme version of the problem of guarding a set of possible polygonalizations that are consistent with a given set of sample points that are the polygon vertices: In the universal setting, we require that the guards are a rich enough set to achieve visibility coverage for all possible polygonalizations. Another variant studied here is the k-universal guarding problem in which the guards must perform visibility coverage for a set of k different polygonalizations of the input points. In the full version of the paper [10], we study the case in which guards are required to be placed at non-convex hull points of S, or at points of a regular rectangular grid
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