TY - JOUR

T1 - A Constant-Factor Approximation Algorithm for Vertex Guarding a WV-Polygon

AU - Ashur, Stav

AU - Filtser, Omrit

AU - Katz, Matthew J.

N1 - Publisher Copyright:
© 2021, Springer Nature Switzerland AG.

PY - 2020

Y1 - 2020

N2 - The problem of vertex guarding a simple polygon was first studied by Subir K. Ghosh (1987), who presented a polynomial-time O(log n) -approximation algorithm for placing as few guards as possible at vertices of a simple n-gon P, such that every point in P is visible to at least one of the guards. Ghosh also conjectured that this problem admits a polynomial-time algorithm with constant approximation ratio. Due to the centrality of guarding problems in the field of computational geometry, much effort has been invested throughout the years in trying to resolve this conjecture. Despite some progress (surveyed below), the conjecture remains unresolved to date. In this paper, we confirm the conjecture for the important case of weakly visible polygons, by presenting a (2 + ε) -approximation algorithm for guarding such a polygon using vertex guards. A simple polygon P is weakly visible if it has an edge e, such that every point in P is visible from some point on e. We also present a (2 + ε) -approximation algorithm for guarding a weakly visible polygon P, where guards may be placed anywhere on P’s boundary (except in the interior of the edge e). Finally, we present an O(1)-approximation algorithm for vertex guarding a polygon P that is weakly visible from a chord. Our algorithms are based on an in-depth analysis of the geometric properties of the regions that remain unguarded after placing guards at the vertices to guard the polygon’s boundary. Finally, our algorithms may become useful as part of the grand attempt of Bhattacharya et al. to prove the original conjecture, as their approach is based on partitioning the underlying simple polygon into a hierarchy of weakly visible polygons.

AB - The problem of vertex guarding a simple polygon was first studied by Subir K. Ghosh (1987), who presented a polynomial-time O(log n) -approximation algorithm for placing as few guards as possible at vertices of a simple n-gon P, such that every point in P is visible to at least one of the guards. Ghosh also conjectured that this problem admits a polynomial-time algorithm with constant approximation ratio. Due to the centrality of guarding problems in the field of computational geometry, much effort has been invested throughout the years in trying to resolve this conjecture. Despite some progress (surveyed below), the conjecture remains unresolved to date. In this paper, we confirm the conjecture for the important case of weakly visible polygons, by presenting a (2 + ε) -approximation algorithm for guarding such a polygon using vertex guards. A simple polygon P is weakly visible if it has an edge e, such that every point in P is visible from some point on e. We also present a (2 + ε) -approximation algorithm for guarding a weakly visible polygon P, where guards may be placed anywhere on P’s boundary (except in the interior of the edge e). Finally, we present an O(1)-approximation algorithm for vertex guarding a polygon P that is weakly visible from a chord. Our algorithms are based on an in-depth analysis of the geometric properties of the regions that remain unguarded after placing guards at the vertices to guard the polygon’s boundary. Finally, our algorithms may become useful as part of the grand attempt of Bhattacharya et al. to prove the original conjecture, as their approach is based on partitioning the underlying simple polygon into a hierarchy of weakly visible polygons.

KW - Approximation algorithms

KW - Art gallery problems

KW - Geometric optimization

KW - Visibility

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SN - 1611-3349

SP - 81

EP - 96

JO - Journal on Computational Geometry

JF - Journal on Computational Geometry

T2 - 18th International Workshop on Approximation and Online Algorithms, WAOA 2019

Y2 - 9 September 2020 through 10 September 2020

ER -