Watchman Routes for Lines and Segments

Dumitrescu, Adrian; Mitchell, Joseph S. B.; Żyliński, Paweł

doi:10.1007/978-3-642-31155-0_4

Cited by 5 publications

(10 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In much the same way that we adapted the proof in [9] of the NP-hardness of the shortest watchman route problem for orthogonal line segments to yield a proof of NP-hardness of MGTS (Theorem 1), we can adapt the proof in [9] of the NP-hardness of the shortest watchman route for an arrangement of lines in 3-space to yield a proof of the NP-hardness of the problem of finding a minimum guarding tree for a connected arrangement of lines in 3-space. To this end we use a reduction from the rectilinear Steiner tree problem [11] and obtain the following result.…”

Section: Resultsmentioning

confidence: 99%

“…In the minimum watchman route problem, the objective is the same as in MGTL or MGTS, but the guarding network is restricted to form a route (closed tour). This variant has been studied in [9,29,30,20]. Xu and Brass [29,30] proved the NP-hardness of this variant for axis-parallel line segments, which was later reproved in a simpler way by Dumitrescu et al [9]; in the latter paper [9], the minimum watchman route problem for lines is shown to be polynomially tractable.…”

Section: The Minimum Guarding Tree Problem For Segments (Mgts)mentioning

confidence: 99%

“…This variant has been studied in [9,29,30,20]. Xu and Brass [29,30] proved the NP-hardness of this variant for axis-parallel line segments, which was later reproved in a simpler way by Dumitrescu et al [9]; in the latter paper [9], the minimum watchman route problem for lines is shown to be polynomially tractable. Recently, Mitchell [20] revisited the minimum watchman route problem and gave a polynomial time algorithm for half-lines and a O(log 2 n)-approximation algorithm for line segments (and more generally for the watchman route problem in polygons with holes).…”

Section: The Minimum Guarding Tree Problem For Segments (Mgts)mentioning

confidence: 99%

“…Our simpler alternative proof is a reduction from rectilinear Steiner tree (RST). The same construction was used in the NP-hardness proof of the minimum watchman route problem for orthogonal line segments in [9].…”

Section: Line Segments In the Plane: Np-hardnessmentioning

confidence: 99%

“…Let L be a connected arrangement of n lines in the plane and let L ′ ⊆ L be a non-empty subset of L. In [9], the authors provide an algorithm for computing a shortest guarding route (i.e., closed tour) R opt for L ′ in O(n 8 ) time. Some edges in R opt may be traversed twice; we take all edges in R opt and remove "multiplicities", if any; the resulting graph is…”

Section: Theorem 2 There Exists a Ratio 2 Approximation Algorithm Fomentioning

confidence: 99%

See 4 more Smart Citations

The Minimum Guarding Tree Problem

Dumitrescu

Mitchell

Żyliński

2014

Discrete Math. Algorithm. Appl.

Self Cite

View full text Add to dashboard Cite

Given a set L of non-parallel lines in the plane and a nonempty subset L ′ ⊆ L, a guarding tree for L ′ is a tree contained in the union of the lines in L such that if a mobile guard (agent) runs on the edges of the tree, all lines in L ′ are visited by the guard. Similarly, given a connected arrangement S of line segments in the plane and a nonempty subset S ′ ⊆ S, we define a guarding tree for S ′ . The minimum guarding tree problem for a given set of lines or line segments is to find a minimum-length guarding tree for the input set.We provide a simple alternative (to [29]) proof of NP-hardness of the problem of finding a guarding tree of minimum length for a set of orthogonal (axis-parallel) line segments in the plane. Then, we present two approximation algorithms with factors 2 and 3.98, respectively, for computing a minimum guarding tree for a subset of a set of n arbitrary non-parallel lines in the plane; their running times are O(n 8 ) and O(n 6 log n), respectively. Finally, we show that this problem is NP-hard for lines in 3-space.

show abstract

Section: Resultsmentioning

confidence: 99%

Section: The Minimum Guarding Tree Problem For Segments (Mgts)mentioning

confidence: 99%

Section: The Minimum Guarding Tree Problem For Segments (Mgts)mentioning

confidence: 99%

Section: Line Segments In the Plane: Np-hardnessmentioning

confidence: 99%

Section: Theorem 2 There Exists a Ratio 2 Approximation Algorithm Fomentioning

confidence: 99%

See 3 more Smart Citations

The Minimum Guarding Tree Problem

Dumitrescu

Mitchell

Żyliński

2014

Discrete Math. Algorithm. Appl.

Self Cite

View full text Add to dashboard Cite

show abstract

The searchlight problem for road networks

Dereniowski

Ono

Suzuki

et al. 2015

Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

We consider the problem of searching for a mobile intruder hiding in a road network given as the union of two or more lines, or two or more line segments, in the plane. Some of the intersections of the road network are occupied by stationary guards equipped with a number of searchlights, each of which can emit a single ray of light in any direction along the lines (or line segments) it is on. The goal is to detect the intruder, that is, to illuminate its location. Guards may alter the direction in which they aim a searchlight, but need to switch it off for some finite time interval to effect the change. In contrast, the intruder may move with arbitrary speed along the network (but cannot pass guards) and exploit this time interval to recontaminate previously illuminated sections of the network. For various classes of road networks characterized by the number n of lines (or line segments) comprising it and the number g (≤ n − 1) of possible locations of guards (fixed in advance and guaranteed to give complete coverage), we present several upper and lower bounds on the worst-case number of searchlights, each placed at one of the guard positions, required to successfully search a given road network. In particular, we prove the following results:1. min{2g − 1, n − 2} searchlights are sometimes necessary and min{ 7 3 g, n} − 1 are always sufficient for searching a road network given as the union of n lines;2. (g · log n g ) searchlights are sometimes necessary and O (g 2 · log n) searchlights are always sufficient for searching a road network given as the union of n line segments, and 3. at most one searchlight per guard position, and hence a total of at most g searchlights, is always sufficient for searching a road network given as the union of axis-aligned lines or line segments. 29 The proofs of the upper bounds induce algorithms for generating a search schedule for detecting the intruder using the claimed number of searchlights.

show abstract