## Terrain Guarding is NP-Hard (2009)

Citations: | 1 - 0 self |

### BibTeX

@MISC{King09terrainguarding,

author = {James King and Erik Krohn},

title = {Terrain Guarding is NP-Hard},

year = {2009}

}

### OpenURL

### Abstract

A set G of points on a 1.5-dimensional terrain, also known as an x-monotone polygonal chain, is said to guard the terrain if every point on the terrain is seen by a point in G. Two points on the terrain see each other if and only if the line segment between them is never strictly below the terrain. The minimum terrain guarding problem asks for a minimum guarding set for the given input terrain. Using a reduction from PLANAR 3-SAT we prove that the decision version of this problem is NP-hard. This solves a significant open problem and complements recent positive approximability results for the optimization problem. 1

### Citations

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Citation Context ...s ensure that, for each variable other than x1 and xn, the variable’s lane is actually bounded by two ‘roadblocks’. 2.5. Running time. A planar embedding of a planar graph can be found in linear time =-=[11, 16]-=-. The other tasks involved in constructing TΦ from Φ can be performed trivially in polynomial time. 2.6. Truth assignment propagation with clause evaluation. We can think of the variable assignment as... |

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Citation Context ...et of the vertices that see the entire polygon. The point guarding version allows guards to be placed anywhere inside the polygon. The art gallery problem was shown to be NPcomplete by Lee and Lin in =-=[13]-=-. Along with being NPcomplete, the art gallery problem was shown to be APXhard in [6]. This means that there exists a constant ɛ > 0 such that no polynomial time algorithm can guarantee an approximati... |

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Citation Context ...er the last decade. The first constant factor approximation for the terrain guarding problem was shown by BenMoshe et al. in [1]. Clarkson and Varadarajan also give a constant factor approximation in =-=[3]-=-. A 4approximation was proposed by King in [11] but further analysis increased the approximation factor to 5. A 4-approximation was given by Elbassioni et al. in [7]. Recently a PTAS was given by Gibs... |

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Citation Context ... the set of clauses on the interior (resp. exterior) of the variable cycle. Let n be the number of variables. The most convenient way for us to now visualize Φ is that given by Knuth and Raghunathan1 =-=[12]-=- 1 The layout described by Knuth and Raghunathan actually has the variables on a horizontal line, but having them on a verticalC1 C2 C3 x1 x2 x3 x4 x5 C4 C5 Figure 1: An instance Φ with three-legged ... |

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Citation Context ...h that no polynomial time algorithm can guarantee an approximation ratio of 1 + ɛ unless P = NP . Ghosh provides a O(log n)-approximation for the problem of vertex guarding an n-vertex simple polygon =-=[8]-=-. The point guarding problem seems to be much harder than the vertex guarding problem and precious little is known about it [5]. A restricted version of the point guardingproblem where the polygon is... |

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Citation Context ...n nodes ensure that, for each variable other than x1 and xn, the variable’s lane is actually bounded by two ‘roadblocks’. Running time A planar embedding of a planar graph can be found in linear time =-=[10, 15]-=-. The other tasks involved in constructing TΦ from Φ can be performed trivially in polynomial time. C1 C2 C3 x1 x2 x3 x4 x5 C4 D1 C5 Truth assignment propagation with clause evaluation We can think of... |

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Citation Context ...hard; reductions from restricted variants are sometimes far simpler than reductions from general 3SAT. In particular, PLANAR 3-SAT is often used to prove NP-hardness of geometric problems (see, e.g., =-=[16]-=-). We adapt the following definition from Mulzer and Rote [16]. Definition 1. (PLANAR 3-SAT) Let Φ be a Boolean formula in 3-CNF. The formula graph of Φ, G(Φ), has one variable-vertex vx for each vari... |

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Citation Context ... lights or security cameras. Other applications include finding a configuration for line-of-sight transmission networks for radio broadcasting, cellular telephony and other communication technologies =-=[1]-=-. The complexity of terrain guarding has been an open problem of interest since 1995, when an NPcompleteness proof was proposed but never completed by Chen et al. [2]. With the problem’s hardness stro... |

17 | Inapproximability Results for Guarding Polygons without Holes
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Citation Context ...ds to be placed anywhere inside the polygon. The art gallery problem was shown to be NPcomplete by Lee and Lin in [13]. Along with being NPcomplete, the art gallery problem was shown to be APXhard in =-=[6]-=-. This means that there exists a constant ɛ > 0 such that no polynomial time algorithm can guarantee an approximation ratio of 1 + ɛ unless P = NP . Ghosh provides a O(log n)-approximation for the pro... |

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Citation Context ...onstant ɛ > 0 such that no polynomial time algorithm can guarantee an approximation ratio of 1 + ɛ unless P = NP. The best approximation guarantee attained for guarding a simple polygon is O(log OPT) =-=[9, 5, 6]-=-, though for the point guarding problem there is still no fully polynomial algorithm that guards the polygon with certainty. The point guarding problem seems to more difficult, though than the vertex ... |

9 |
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Citation Context ... other communication technologies [1]. The complexity of terrain guarding has been an open problem of interest since 1995, when an NPcompleteness proof was proposed but never completed by Chen et al. =-=[2]-=-. With the problem’s hardness strongly suspected but not known, a series of approximation algorithms have been developed over the last decade. The first constant factor approximation for the terrain g... |

8 |
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Citation Context ...arding problem and precious little is known about it [5]. A restricted version of the point guardingproblem where the polygon is x-monotone has been shown to have an O(1)-approximation by Nilsson in =-=[17]-=-. Based on his result Nilsson also provides a O(OP T 2 ) approximation for rectilinear polygons. Straightforward attempts to show NP-hardness for the terrain guarding problem run up against the large ... |

5 | A 4-approximation algorithm for guarding 1.5-dimensional terrains
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(Show Context)
Citation Context ...approximation for the terrain guarding problem was shown by BenMoshe et al. in [1]. Clarkson and Varadarajan also give a constant factor approximation in [3]. A 4approximation was proposed by King in =-=[11]-=- but further analysis increased the approximation factor to 5. A 4-approximation was given by Elbassioni et al. in [7]. Recently a PTAS was given by Gibson et al. in [9]. With the knowledge that the p... |

4 |
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Citation Context ..., an exact polynomial time algorithm is not possible unless P = NP. The NP-hardness result is shown for the standard discrete and continuous variants of the problem. According to Demaine and O’Rourke =-=[4]-=-, the complexity of the terrain guarding problem was posed by Ben-Moshe. We quote from [4]: What is the complexity of computing the guard set of minimum size for a given x-monotone chain in the plane?... |

3 |
Open problems: Open problems from CCCG 2005
- Demaine, O’Rourke
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(Show Context)
Citation Context ..., an exact polynomial time algorithm is not possible unless P = NP. The NP-hardness result is shown for the standard discrete and continuous variants of the problem. According to Demaine and O’Rourke =-=[4]-=-, the complexity of the terrain guarding problem was posed by Ben-Moshe. We quote from [4]: What is the complexity of computing the guard set of minimum size for a given xmonotone chain in the plane? ... |

2 |
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(Show Context)
Citation Context ...roximation for the problem of vertex guarding an n-vertex simple polygon [8]. The point guarding problem seems to be much harder than the vertex guarding problem and precious little is known about it =-=[5]-=-. A restricted version of the point guardingproblem where the polygon is x-monotone has been shown to have an O(1)-approximation by Nilsson in [17]. Based on his result Nilsson also provides a O(OP T... |

2 | Improved approximations for guarding 1.5dimensional terrains
- Elbassioni, Matijević, et al.
(Show Context)
Citation Context ...a constant factor approximation in [3]. A 4approximation was proposed by King in [11] but further analysis increased the approximation factor to 5. A 4-approximation was given by Elbassioni et al. in =-=[7]-=-. Recently a PTAS was given by Gibson et al. in [9]. With the knowledge that the problem is not APXcomplete, it is of even greater interest whether or not it is NP-complete, and this has been reiterat... |

1 |
An Approximation Scheme for Terrain Guarding
- Gibson, Kanade, et al.
- 2009
(Show Context)
Citation Context ...ation was proposed by King in [11] but further analysis increased the approximation factor to 5. A 4-approximation was given by Elbassioni et al. in [7]. Recently a PTAS was given by Gibson et al. in =-=[9]-=-. With the knowledge that the problem is not APXcomplete, it is of even greater interest whether or not it is NP-complete, and this has been reiterated with each approximation algorithm developed. The... |