Approximating Domination on Intersection Graphs of Paths on a Grid

Mehrabi, Saeed

doi:10.1007/978-3-319-89441-6_7

Cited by 8 publications

(8 citation statements)

References 26 publications

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“…To this end, we give an L-reduction from the Minimum Vertex Cover (MVC) problem on graphs with maximum-degree three to this variant of the LSC problem. Our reduction is inspired by the construction of Mehrabi [10]. As a reminder, we first give a formal definition of L-reduction [12], which is one of the gap-preserving reductions.…”

Section: Apx-hardnessmentioning

confidence: 99%

Approximability of Covering Cells with Line Segments

Carmi

Maheshwari

Mehrabi

et al. 2018

Combinatorial Optimization and Applications

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In COCOA 2015, Korman et al. studied the following geometric covering problem: given a set S of n line segments in the plane, find a minimum number of line segments such that every cell in the arrangement of the line segments is covered. Here, a line segment s covers a cell f if s is incident to f . The problem was shown to be NP-hard, even if the line segments in S are axis-parallel, and it remains NP-hard when the goal is cover the "rectangular" cells (i.e., cells that are defined by exactly four axis-parallel line segments).In this paper, we consider the approximability of the problem. We first give a PTAS for the problem when the line segments in S are in any orientation, but we can only select the covering line segments from one orientation. Then, we show that when the goal is to cover the rectangular cells using line segments from both horizontal and vertical line segments, then the problem is APX-hard. We also consider the parameterized complexity of the problem and prove that the problem is FPT when parameterized by the size of an optimal solution. Our FPT algorithm works when the line segments in S have two orientations and the goal is to cover all cells, complementing that of Korman et al. [9] in which the goal is to cover the "rectangular" cells.

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Section: Apx-hardnessmentioning

confidence: 99%

Approximability of Covering Cells with Line Segments

Carmi

Maheshwari

Mehrabi

et al. 2018

Combinatorial Optimization and Applications

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“…Mehrabi [16] gave an -net based O(1)-approximation algorithm for the MDS problem on one-string B 1 -VPG graphs (graphs with B 1 -VPG representation where two curves intersect at most once). Bandyapadhyay et al [2] proved APX-hardness for the MDS problem on a special class of B 1 -VPG graphs, namely verticallystabbed-L graph (defined below) which was originally introduced by McGuinness [15].…”

Section: Introductionmentioning

confidence: 99%

“…Many researchers have studied the MDS problem on circle graphs [5,8,9]. Since all vertically-stabbed-Lgraphs are also one-string B 1 -VPG graphs, there is a O(1)-approximation algorithm for the MDS problem on vertically-stabbed-L graphs (due to Mehrabi [16]). In this paper, we prove the following theorems.…”

Section: Introductionmentioning

confidence: 99%

Approximating Minimum Dominating Set on String Graphs

Chakraborty

Das

Mukherjee

2019

Lecture Notes in Computer Science

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In this paper, we give approximation algorithms for the Minimum Dominating Set (MDS) problem on string graphs and its subclasses. A path is a simple curve made up of alternating horizontal and vertical line segments. A k-bend path is a path made up of at most k + 1 line segments. An L-path is a 1-bend path having the shape 'L'. A vertically-stabbed-L graph is an intersection graph of L-paths intersecting a common vertical line. We give a polynomial time 8-approximation algorithm for MDS problem on vertically-stabbed-L graphs whose APX-hardness was shown by Bandyapadhyay et al. (MFCS, 2018). To prove the above result, we needed to study the Stabbing segments with rays (SSR) problem introduced by Katz et al. (Comput. Geom. 2005). In the SSR problem, the input is a set of (disjoint) leftwarddirected rays, and a set of (disjoint) vertical segments. The objective is to select a minimum number of rays that intersect all vertical segments. We give a O((n+m) log(n+m))-time 2-approximation algorithm for the SSR problem where n and m are the number of rays and segments in the input. A unit k-bend path is a k-bend path whose segments are of unit length. A graph is a unit B k -VPG graph if it is an intersection graph of unit k-bend paths. Any string graph is a unit-B k -VPG graph for some finite k. Using our result on SSR-problem, we give a polynomial time O(k 4 )-approximation algorithm for MDS problem on unit B k -VPG graphs for k ≥ 0.

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“…These graphs and their relatives have been studied extensively in terms of recognition problems (e.g., see [19,14,11,4]). Recently, there has been an increasing attention on studying optimization problems on these graphs; see [5,27,6,28] and the references therein. For the MIS problem, it is known that the problem is NP-complete on B k -VPG graphs even when k = 1 [25], and the previously best-known approximation algorithms have factor 4 • log n [6,28].…”

Section: Introductionmentioning

confidence: 99%

Computing Maximum Independent Set on Outerstring Graphs and Their Relatives

Bose

Carmi

Keil

et al. 2019

Lecture Notes in Computer Science

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A graph G with n vertices is called an outerstring graph if it has an intersection representation of a set of n curves inside a disk such that one endpoint of every curve is attached to the boundary of the disk. Given an outerstring graph representation, the Maximum Independent Set (MIS) problem of the underlying graph can be solved in O(s 3 ) time, where s is the number of segments in the representation (Keil et al., Comput. Geom., 60:19-25, 2017). If the strings are of constant size (e.g., line segments, L-shapes, etc.), then the algorithm takes O(n 3 ) time.In this paper, we examine the fine-grained complexity of the MIS problem on some well-known outerstring representations. We show that solving the MIS problem on grounded segment and grounded square-L representations is at least as hard as solving MIS on circle graph representations. Note that no O(n 2−δ )-time algorithm, δ > 0, is known for the MIS problem on circle graphs. For the grounded string representations where the strings are y-monotone simple polygonal paths of constant length with segments at integral coordinates, we solve MIS in O(n 2 ) time and show this to be the best possible under the strong exponential time hypothesis (SETH). For the intersection graph of n L-shapes in the plane, we give a (4 · log OPT)-approximation algorithm for MIS (where OPT denotes the size of an optimal solution), improving the previously best-known (4 · log n)-approximation algorithm of Biedl and Derka (WADS 2017).

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Approximating Domination on Intersection Graphs of Paths on a Grid

Cited by 8 publications

References 26 publications

Approximability of Covering Cells with Line Segments

Approximability of Covering Cells with Line Segments

Approximating Minimum Dominating Set on String Graphs

Computing Maximum Independent Set on Outerstring Graphs and Their Relatives

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