Dominating set of rectangles intersecting a straight line

Pandit, Supantha

doi:10.1007/s10878-020-00685-y

Cited by 2 publications

(2 citation statements)

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“…For general graphs, it is known that a greedy algorithm for MDS achieves an O(log |V |)factor approximation and within a constant factor this is the best one can hope for, unless P=NP [27]. As such, the problem has been extensively studied on many subclasses of graphs, one of which is the intersection graphs of geometric objects in the plane [11,22,16,15,23,26].…”

Section: Introductionmentioning

confidence: 99%

“…However, even for simple shapes such as axis-parallel rectangles no sub-logarithmic approximation is known. The only approximation for rectangles we are aware of is due to Erlebach et al [15] who gave an O(c 3 )-approximation on rectangles with aspect-ratio at most c. In fact, the problem is APX-hard [15] on rectangles, and (perhaps surprisingly) the problem is shown to be NP-hard even on diagonal-anchored rectangles [26]; that is, the intersection of every rectangle and a diagonal line with slope -1 is exactly one corner of the rectangle. See Figure 2(a) for an example.…”

Section: Introductionmentioning

confidence: 99%

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Approximating dominating set on intersection graphs of rectangles and L-frames

Bandyapadhyay

Maheshwari

Mehrabi

et al. 2019

Computational Geometry

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We consider the Minimum Dominating Set (MDS) problem on the intersection graphs of geometric objects. Even for simple and widely-used geometric objects such as rectangles, no sublogarithmic approximation is known for the problem and (perhaps surprisingly) the problem is NP-hard even when all the rectangles are "anchored" at a diagonal line with slope -1 (Pandit, CCCG 2017). In this paper, we first show that for any > 0, there exists a (2 + )-approximation algorithm for the MDS problem on "diagonal-anchored" rectangles, providing the first O(1)approximation for the problem on a non-trivial subclass of rectangles. It is not hard to see that the MDS problem on "diagonal-anchored" rectangles is the same as the MDS problem on "diagonal-anchored" L-frames: the union of a vertical and a horizontal line segment that share an endpoint. As such, we also obtain a (2 + )-approximation for the problem with "diagonalanchored" L-frames. On the other hand, we show that the problem is APX-hard in case the input L-frames intersect the diagonal, or the horizontal segments of the L-frames intersect a vertical line. However, as we show, the problem is linear-time solvable in case the L-frames intersect a vertical as well as a horizontal line. Finally, we consider the MDS problem in the so-called "edge intersection model" and obtain a number of results, answering two questions posed by Mehrabi (WAOA 2017).

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Approximating dominating set on intersection graphs of rectangles and L-frames

Bandyapadhyay

Maheshwari

Mehrabi

et al. 2019

Computational Geometry

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A Survey on Variant Domination Problems in Geometric Intersection Graphs

Xu,

Wang,

Yang

2024

Parallel Process. Lett.

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Given a graph [Formula: see text] with vertex set [Formula: see text], a subset [Formula: see text] of [Formula: see text] is called a dominating set of [Formula: see text] if every vertex is either in [Formula: see text] or is adjacent to a vertex in [Formula: see text]. There are a lot of variants for dominating sets, such as connected dominating sets, total dominating sets, total restricted dominating sets, secure (connected, total) dominating sets, etc. Geometric intersection graphs are graphs for which there is a bijection [Formula: see text] between the vertices and a set [Formula: see text] of geometric objects (for example, disks, rectangles, etc.) such that there is an edge between two vertices [Formula: see text] and [Formula: see text] if and only if the objects [Formula: see text] and [Formula: see text] intersect. In this paper, we offer a survey about complexity and algorithmic results on variant domination problems in geometric intersection graphs.

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Dominating set of rectangles intersecting a straight line

Cited by 2 publications

References 18 publications

Approximating dominating set on intersection graphs of rectangles and L-frames

Approximating dominating set on intersection graphs of rectangles and L-frames

A Survey on Variant Domination Problems in Geometric Intersection Graphs

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