The Inapproximability of Illuminating Polygons by α-Floodlights

We consider variants of the art gallery problem where guard visibility is limited to a certain angular aperture α. We show that the problem is NP-hard even when guards can be located in the interior of the polygon. We then proceed to prove that both this problem and its vertex variant, where guard placement is restricted to the vertices of the polygon, are APX-hard. We observe that earlier constructions for such results in art gallery problems with 360◦ guards, usually required them to cover few specific elements. We exploit this by carefully updating the construction to replace 360◦ guards with α-floodlights. Similar transformations may be applicable to other constructions in traditional art gallery theorems, which is of independent interest.

[1]  Sariel Har-Peled,et al.  Guarding galleries and terrains , 2002, Inf. Process. Lett..

[2]  Joseph C. Culberson,et al.  Covering polygons is hard , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[3]  David Eppstein,et al.  Guard placement for efficient point-in-polygon proofs , 2007, SCG '07.

[4]  Euripides Markou,et al.  Maximizing the guarded boundary of an Art Gallery is APX-complete , 2003, Comput. Geom..

[5]  Atri Rudra,et al.  Floodlight illumination of infinite wedges , 2010, Comput. Geom..

[6]  Bengt J. Nilsson,et al.  Guarding lines and 2-link polygons is apx-hard , 2001, CCCG.

[7]  T. C. Shermer,et al.  Recent results in art galleries (geometry) , 1992, Proc. IEEE.

[8]  Leonidas J. Guibas,et al.  The Floodlight Problem , 1997, Int. J. Comput. Geom. Appl..

[9]  D. T. Lee,et al.  Computational complexity of art gallery problems , 1986, IEEE Trans. Inf. Theory.

[10]  Ján Manuch,et al.  Monitoring the Plane with Rotating Radars , 2015, Graphs Comb..

[11]  J. O'Rourke.  Art gallery theorems and algorithms , 1987 .

[12]  Joseph S. B. Mitchell,et al.  Locating Guards for Visibility Coverage of Polygons , 2007, Int. J. Comput. Geom. Appl..

[13]  Jorge Urrutia,et al.  Optimal floodlight illumination of stages , 1998, SCG '98.

[14]  Stephan Eidenbenz,et al.  Inapproximability Results for Guarding Polygons and Terrains , 2001, Algorithmica.

[15]  Andrea Bottino,et al.  A nearly optimal algorithm for covering the interior of an Art Gallery , 2011, Pattern Recognit..

[16]  Stephan Eidenbenz,et al.  An Approximation Algorithm for Minimum Convex Cover with Logarithmic Performance Guarantee , 2003, SIAM J. Comput..

[17]  David Glasser,et al.  The Complexity of Illuminating Polygons by alpha-flood-lights , 1996, CCCG.

[18]  Jorge Urrutia,et al.  Optimal Floodlight Illumination of Orthogonal Art Galleries , 1994, Canadian Conference on Computational Geometry.

[19]  James King.  Fast vertex guarding for polygons with and without holes , 2013, Comput. Geom..

[20]  Hans-Dietrich Hecker,et al.  Minimizing the Size of Vertexlights in Simple Polygons , 2002, Math. Log. Q..

[21]  Mitsuo Yokoyama,et al.  NP-Completeness of Stage Illumination Problems , 1998, JCDCG.

[22]  Ileana Streinu,et al.  Illumination by floodlights , 1998, Comput. Geom..

[23]  E. M ARKOU.  Approximating Visibility Problems within a Constant , 2006 .