Finding the Minimum Number of Face Guards is NP-Hard

Iwamoto, Chuzo; Kitagaki, Yusuke; Morita, Kenichi

doi:10.1587/transinf.e95.d.2716

Cited by 5 publications

(6 citation statements)

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“…We also obtain the same result for (non-triangulated) terrains. This adds to the result of [10], which states that minimizing closed face guards is NP-hard in triangulated terrains. We also briefly discuss the membership in NP of the minimization problem, pointing out some difficulties in applying previously known techniques.…”

Section: Open Face Guardsmentioning

confidence: 95%

“…We also show that the same hardness of approximation result holds for non-triangulated terrains. Recall that, in [10], Iwamoto et al proved that minimizing closed face guards in triangulated terrains is NP-hard. Thus, we improve on their result in the case of non-triangulated terrains, while also extending it to open face guards.…”

Section: Minimizing Face Guards 41 Hardness Of Approximationmentioning

confidence: 99%

“…A terrain whose bounded faces are triangles is called a triangulated terrain. These special terrains are studied in [9,10,11], where it is shown that computing the minimum number of closed face guards in a given triangulated terrain is NP-hard. Here we strengthen this result by showing that such a minimum is even NP-hard to approximate within a logarithmic factor, for both open and closed face guards, provided that terrains are not necessarily triangulated.…”

Section: Minimizing Face Guards 41 Hardness Of Approximationmentioning

confidence: 99%

“…Subsequently, face guards have been studied to some extent also in the case of polyhedral terrains. In [9,11] a tight bound is obtained, and in [10] it is proven that minimizing face guards in triangulated terrains is NPhard. However, since these results apply to terrains, they have no direct implications on the problem of face-guarding polyhedral enclosures.…”

Section: Introductionmentioning

confidence: 99%

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Face-guarding polyhedra

Viglietta

2014

Computational Geometry

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We study the Art Gallery Problem for face guards in polyhedral environments. The problem can be informally stated as: how many (not necessarily convex) windows should we place on the external walls of a dark building, in order to completely illuminate its interior?We consider both closed and open face guards (i.e., faces with or without their boundary), and we study several classes of polyhedra, including orthogonal polyhedra, 4-oriented polyhedra, and 2-reflex orthostacks.We give upper and lower bounds on the minimum number of faces required to guard the interior of a given polyhedron in each of these classes, in terms of the total number of its faces, f . In several cases our bounds are tight: f /6 open face guards for orthogonal polyhedra and 2-reflex orthostacks, and f /4 open face guards for 4-oriented polyhedra. Additionally, for closed face guards in 2-reflex orthostacks, we give a lower bound of (f +3)/9 and an upper bound of (f +1)/7 .Then we show that it is NP-hard to approximate the minimum number of (closed or open) face guards within a factor of Ω(log f ), even for polyhedra that are orthogonal and simply connected. We also obtain the same hardness results for polyhedral terrains.Along the way we discuss some applications, arguing that face guards are not a reasonable model for guards patrolling on the surface of a polyhedron.

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Section: Open Face Guardsmentioning

confidence: 95%

Section: Minimizing Face Guards 41 Hardness Of Approximationmentioning

confidence: 99%

Section: Minimizing Face Guards 41 Hardness Of Approximationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Face-guarding polyhedra

Viglietta

2014

Computational Geometry

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“…As noted above, covering a 3-dimensional polyhedron by vertex guards may be unfeasible. There are some initial results on the minimum number of edge guards [2,5,13,27] and face guards [14,15,24,26], under the notion of weak visibility: An edge or a face f sees a point p if some point s ∈ f sees the point p. The problem formulation for edge and face guards can be further refined depending on whether (topologically) open or closed edges and faces are allowed. However, none of the current bounds is known to be tight.…”

Section: Introductionmentioning

confidence: 99%