A Constant–Factor Approximation Algorithm for Red–Blue Set Cover with Unit Disks

The main contribution of this paper is the first constant factor approximation algorithm for red-blue set cover problem with unit disks. To achieve this, we first give a polynomial time algorithm for line-separable red-blue set cover problem with unit disks. We next obtain a factor 2 approximation a...

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Veröffentlicht in:	Algorithmica Jg. 85; H. 1; S. 100 - 132
Hauptverfasser:	Madireddy, Raghunath Reddy, Mudgal, Apurva
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.01.2023 Springer Nature B.V
Schlagworte:	Algorithm Analysis and Problem Complexity Algorithms Approximation Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Disks Mathematical analysis Mathematics of Computing Polynomials Strip Theory of Computation Sweep-line method Unit disks Strip-separable Line-separable Red–blue set cover
ISSN:	0178-4617, 1432-0541
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Abstract	The main contribution of this paper is the first constant factor approximation algorithm for red-blue set cover problem with unit disks. To achieve this, we first give a polynomial time algorithm for line-separable red-blue set cover problem with unit disks. We next obtain a factor 2 approximation algorithm for strip-separable red-blue set cover problem with unit disks. Finally, we obtain a constant factor approximation algorithm for red-blue set cover problem with unit disks by combining our algorithm for the strip-separable problem with the results of Ambühl et al. [ 1 ]. Our methods involve a novel decomposition of the optimal solution to line-separable problem into blocks with special structure and extensions of the sweep-line technique of Erlebach and van Leeuwen [ 9 ].
AbstractList	The main contribution of this paper is the first constant factor approximation algorithm for red-blue set cover problem with unit disks. To achieve this, we first give a polynomial time algorithm for line-separable red-blue set cover problem with unit disks. We next obtain a factor 2 approximation algorithm for strip-separable red-blue set cover problem with unit disks. Finally, we obtain a constant factor approximation algorithm for red-blue set cover problem with unit disks by combining our algorithm for the strip-separable problem with the results of Ambühl et al. [1]. Our methods involve a novel decomposition of the optimal solution to line-separable problem into blocks with special structure and extensions of the sweep-line technique of Erlebach and van Leeuwen [9]. The main contribution of this paper is the first constant factor approximation algorithm for red-blue set cover problem with unit disks. To achieve this, we first give a polynomial time algorithm for line-separable red-blue set cover problem with unit disks. We next obtain a factor 2 approximation algorithm for strip-separable red-blue set cover problem with unit disks. Finally, we obtain a constant factor approximation algorithm for red-blue set cover problem with unit disks by combining our algorithm for the strip-separable problem with the results of Ambühl et al. [ 1 ]. Our methods involve a novel decomposition of the optimal solution to line-separable problem into blocks with special structure and extensions of the sweep-line technique of Erlebach and van Leeuwen [ 9 ].
Author	Madireddy, Raghunath Reddy Mudgal, Apurva
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Keywords	Sweep-line method Unit disks Strip-separable Line-separable Red–blue set cover
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References	ChanTMHuNGeometric Red Blue Set Cover for Unit Squares and Related ProblemsComput. Geom.201548538038510.1016/j.comgeo.2014.12.0051314.65029 FowlerRJPatersonMSTanimotoSLOptimal Packing and Covering in the Plane are NP-CompleteInf. Process. Lett.198112313313710.1016/0020-0190(81)90111-30469.68053 BasappaMAcharyyaRDasGKUnit Disk Cover Problem in 2DJ. Discrete Algorithms20153319320110.1016/j.jda.2015.05.0021337.68264 ErlebachTvan LeeuwenEJPTAS for Weighted Set Cover on Unit Squares. APPROX/RANDOM’102010BerlinSpringer1661771304.68214 ClaudeFDorrigivRDurocherSFraserRLópez-OrtizASalingerAPractical Discrete Unit Disk Cover Using an Exact Line-Separable Algorithm2009BerlinSpringer45541272.68150 LiJJinYA PTAS for the Weighted Unit Disk Cover Problem2015BerlinSpringer8989091440.68335 MustafaNHRaySImproved Results on Geometric Hitting Set ProblemsDiscrete Comput. Geom.201044488389510.1007/s00454-010-9285-91207.68420 DasGKFraserRLòpez-OrtizANickersonBGOn the Discrete Unit Disk Cover Problem2011BerlinSpringer1461571317.68273 Carr, R.D., Doddi, S., Konjevod, G., Marathe, M.: On the Red-blue Set Cover Problem. In: Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’00, pp. 345–353. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2000) FraserRLòpez-OrtizAThe Within-Strip Discrete Unit Disk Cover ProblemTheor. Comput. Sci.20176749911510.1016/j.tcs.2017.01.0301370.68297 Ambühl, C., Erlebach, T., Mihalák, M., Nunkesser, M.: Constant-Factor Approximation for Minimum-Weight (Connected) Dominating Sets in Unit Graphs. In: 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX, pp. 3–14 (2006) CălinescuGMăndoiuIIWanPJZelikovskyAZSelecting forwarding neighbors in wireless ad hoc networksMobile Netw. Appl.20049210111110.1023/B:MONE.0000013622.63511.57 CarmiPKatzMJLev-TovNCovering Points by Unit Disks of Fixed Location2007BerlinSpringer6446551193.68268 Madireddy, R.R., Mudgal, A.: A constant-factor approximation algorithm for red-blue set cover with unit disks. In: 18th International Workshop on Approximation and Online Algorithms WAOA 2020, Lecture Notes in Computer Science, vol. 12806, pp. 204–219. Springer (2020) P Carmi (1012_CR4) 2007 F Claude (1012_CR7) 2009 GK Das (1012_CR8) 2011 1012_CR5 1012_CR1 M Basappa (1012_CR2) 2015; 33 J Li (1012_CR12) 2015 G Călinescu (1012_CR3) 2004; 9 RJ Fowler (1012_CR10) 1981; 12 TM Chan (1012_CR6) 2015; 48 R Fraser (1012_CR11) 2017; 674 T Erlebach (1012_CR9) 2010 NH Mustafa (1012_CR14) 2010; 44 1012_CR13
References_xml	– reference: ChanTMHuNGeometric Red Blue Set Cover for Unit Squares and Related ProblemsComput. Geom.201548538038510.1016/j.comgeo.2014.12.0051314.65029 – reference: BasappaMAcharyyaRDasGKUnit Disk Cover Problem in 2DJ. Discrete Algorithms20153319320110.1016/j.jda.2015.05.0021337.68264 – reference: Ambühl, C., Erlebach, T., Mihalák, M., Nunkesser, M.: Constant-Factor Approximation for Minimum-Weight (Connected) Dominating Sets in Unit Graphs. In: 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX, pp. 3–14 (2006) – reference: CălinescuGMăndoiuIIWanPJZelikovskyAZSelecting forwarding neighbors in wireless ad hoc networksMobile Netw. Appl.20049210111110.1023/B:MONE.0000013622.63511.57 – reference: FowlerRJPatersonMSTanimotoSLOptimal Packing and Covering in the Plane are NP-CompleteInf. Process. Lett.198112313313710.1016/0020-0190(81)90111-30469.68053 – reference: Madireddy, R.R., Mudgal, A.: A constant-factor approximation algorithm for red-blue set cover with unit disks. In: 18th International Workshop on Approximation and Online Algorithms WAOA 2020, Lecture Notes in Computer Science, vol. 12806, pp. 204–219. Springer (2020) – reference: CarmiPKatzMJLev-TovNCovering Points by Unit Disks of Fixed Location2007BerlinSpringer6446551193.68268 – reference: DasGKFraserRLòpez-OrtizANickersonBGOn the Discrete Unit Disk Cover Problem2011BerlinSpringer1461571317.68273 – reference: ClaudeFDorrigivRDurocherSFraserRLópez-OrtizASalingerAPractical Discrete Unit Disk Cover Using an Exact Line-Separable Algorithm2009BerlinSpringer45541272.68150 – reference: ErlebachTvan LeeuwenEJPTAS for Weighted Set Cover on Unit Squares. APPROX/RANDOM’102010BerlinSpringer1661771304.68214 – reference: MustafaNHRaySImproved Results on Geometric Hitting Set ProblemsDiscrete Comput. Geom.201044488389510.1007/s00454-010-9285-91207.68420 – reference: FraserRLòpez-OrtizAThe Within-Strip Discrete Unit Disk Cover ProblemTheor. Comput. Sci.20176749911510.1016/j.tcs.2017.01.0301370.68297 – reference: LiJJinYA PTAS for the Weighted Unit Disk Cover Problem2015BerlinSpringer8989091440.68335 – reference: Carr, R.D., Doddi, S., Konjevod, G., Marathe, M.: On the Red-blue Set Cover Problem. In: Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’00, pp. 345–353. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2000) – volume: 9 start-page: 101 issue: 2 year: 2004 ident: 1012_CR3 publication-title: Mobile Netw. Appl. doi: 10.1023/B:MONE.0000013622.63511.57 – start-page: 898 volume-title: A PTAS for the Weighted Unit Disk Cover Problem year: 2015 ident: 1012_CR12 – start-page: 166 volume-title: PTAS for Weighted Set Cover on Unit Squares. APPROX/RANDOM’10 year: 2010 ident: 1012_CR9 – volume: 44 start-page: 883 issue: 4 year: 2010 ident: 1012_CR14 publication-title: Discrete Comput. Geom. doi: 10.1007/s00454-010-9285-9 – volume: 12 start-page: 133 issue: 3 year: 1981 ident: 1012_CR10 publication-title: Inf. Process. Lett. doi: 10.1016/0020-0190(81)90111-3 – volume: 33 start-page: 193 year: 2015 ident: 1012_CR2 publication-title: J. Discrete Algorithms doi: 10.1016/j.jda.2015.05.002 – volume: 674 start-page: 99 year: 2017 ident: 1012_CR11 publication-title: Theor. Comput. Sci. doi: 10.1016/j.tcs.2017.01.030 – ident: 1012_CR5 – volume: 48 start-page: 380 issue: 5 year: 2015 ident: 1012_CR6 publication-title: Comput. Geom. doi: 10.1016/j.comgeo.2014.12.005 – ident: 1012_CR1 doi: 10.1007/11830924_3 – start-page: 146 volume-title: On the Discrete Unit Disk Cover Problem year: 2011 ident: 1012_CR8 – ident: 1012_CR13 doi: 10.1007/978-3-030-80879-2_14 – start-page: 45 volume-title: Practical Discrete Unit Disk Cover Using an Exact Line-Separable Algorithm year: 2009 ident: 1012_CR7 – start-page: 644 volume-title: Covering Points by Unit Disks of Fixed Location year: 2007 ident: 1012_CR4
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Approximation Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Disks Mathematical analysis Mathematics of Computing Polynomials Strip Theory of Computation
Title	A Constant–Factor Approximation Algorithm for Red–Blue Set Cover with Unit Disks
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