Geometric Hitting Set for Segments of Few Orientations

Fekete, Sándor P.; Huang, Kan; Mitchell, Joseph S.  B.; Parekh, Ojas; Phillips, Cynthia A.

doi:10.1007/s00224-016-9744-7

Geometric Hitting Set for Segments of Few Orientations

Journal Article · Mon Feb 06 00:00:00 EST 2017 · Theory of Computing Systems

DOI:https://doi.org/10.1007/s00224-016-9744-7· OSTI ID:1464175

Fekete, Sándor P. ^[1]; Huang, Kan ^[2]; ^[2]; Parekh, Ojas ^[3]; Phillips, Cynthia A. ^[3]

Braunschweig Univ. of Technology (Germany)
Stony Brook Univ., NY (United States)
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

In this paper, we study several natural instances of the geometric hitting set problem for input consisting of sets of line segments (and rays, lines) having a small number of distinct slopes. These problems model path monitoring (e.g., on road networks) using the fewest sensors (the “hitting points”). We give approximation algorithms for cases including (i) lines of 3 slopes in the plane, (ii) vertical lines and horizontal segments, (iii) pairs of horizontal/vertical segments. We give hardness and hardness of approximation results for these problems. Finally, we prove that the hitting set problem for vertical lines and horizontal rays is polynomially solvable.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Stony Brook Univ., Stony Brook, NY (United States)

Sponsoring Organization:: National Science Foundation (NSF) (United States); SNL Laboratory Directed Research and Development (LDRD) Program; US-Israel Binational Science Foundation; USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC04-94AL85000

OSTI ID:: 1464175

Report Number(s):: SAND--2016-12946J; 650133

Journal Information:: Theory of Computing Systems, Journal Name: Theory of Computing Systems Journal Issue: 2 Vol. 62; ISSN 1432-4350

Publisher:: SpringerCopyright Statement

Country of Publication:: United States

Language:: English

References (33)

Approximation algorithms for hitting objects with straight lines Hassin, Refael; Megiddo, Nimrod Discrete Applied Mathematics, Vol. 30, Issue 1 https://doi.org/10.1016/0166-218x(91)90011-k	journal	January 1991
Approximation algorithms for a geometric set cover problem Brimkov, Valentin E.; Leach, Andrew; Wu, Jimmy Discrete Applied Mathematics, Vol. 160, Issue 7-8 https://doi.org/10.1016/j.dam.2011.11.023	journal	May 2012
Guarding a set of line segments in the plane Brimkov, Valentin E.; Leach, Andrew; Mastroianni, Michael Theoretical Computer Science, Vol. 412, Issue 15 https://doi.org/10.1016/j.tcs.2010.08.014	journal	March 2011
Approximability issues of guarding a set of segments Brimkov, Valentin E. International Journal of Computer Mathematics, Vol. 90, Issue 8 https://doi.org/10.1080/00207160.2013.775423	journal	August 2013
Approximation schemes for covering and packing problems in image processing and VLSI Hochbaum, Dorit S.; Maass, Wolfgang Journal of the ACM, Vol. 32, Issue 1 https://doi.org/10.1145/2455.214106	journal	January 1985
The NP-Hardness of Covering Points with Lines, Paths and Tours and their Tractability with FPT-Algorithms Heednacram, Apichat Griffith University https://doi.org/10.25904/1912/1694	text	January 2010
Hardness of Set Cover with Intersection 1 Kumar, V. S. Anil; Arya, Sunil; Ramesh, H. Automata, Languages and Programming https://doi.org/10.1007/3-540-45022-X_53	book	January 2000
A 2 1/10-Approximation Algorithm for a Generalization of the Weighted Edge-Dominating Set Problem Carr, Robert; Fujito, Toshihiro; Konjevod, Goran Algorithms - ESA 2000 https://doi.org/10.1007/3-540-45253-2_13	book	January 2000
Constant Ratio Approximation Algorithms for the Rectangle Stabbing Problem and the Rectilinear Partitioning Problem Gaur, Daya Ram; Ibaraki, Toshihide; Krishnamurti, Ramesh Algorithms - ESA 2000 https://doi.org/10.1007/3-540-45253-2_20	book	January 2000
Algorithms for polytope covering and approximation Clarkson, Kenneth L. Lecture Notes in Computer Science https://doi.org/10.1007/3-540-57155-8_252	book	January 1993
Approximation Algorithms for Hitting Triangle-Free Sets of Line Segments Joshi, Anup; Narayanaswamy, N. S. Algorithm Theory – SWAT 2014 https://doi.org/10.1007/978-3-319-08404-6_31	book	January 2014
Parameterized Complexity of Stabbing Rectangles and Squares in the Plane Dom, Michael; Fellows, Michael R.; Rosamond, Frances A. WALCOM: Algorithms and Computation https://doi.org/10.1007/978-3-642-00202-1_26	book	January 2009
Experimental Study on Approximation Algorithms for Guarding Sets of Line Segments Brimkov, Valentin E.; Leach, Andrew; Mastroianni, Michael Advances in Visual Computing https://doi.org/10.1007/978-3-642-17289-2_57	book	January 2010
Almost optimal set covers in finite VC-dimension Brönnimann, H.; Goodrich, M. T. Discrete & Computational Geometry, Vol. 14, Issue 4 https://doi.org/10.1007/BF02570718	journal	December 1995
Covering Things with Things Langerman, Stefan; Morin, Pat Discrete & Computational Geometry, Vol. 33, Issue 4 https://doi.org/10.1007/s00454-004-1108-4	journal	July 2004
Improved Approximation Algorithms for Geometric Set Cover Clarkson, Kenneth L.; Varadarajan, Kasturi Discrete & Computational Geometry, Vol. 37, Issue 1 https://doi.org/10.1007/s00454-006-1273-8	journal	October 2006
Improved Results on Geometric Hitting Set Problems Mustafa, Nabil H.; Ray, Saurabh Discrete & Computational Geometry, Vol. 44, Issue 4 https://doi.org/10.1007/s00454-010-9285-9	journal	September 2010
A Non-linear Lower Bound for Planar Epsilon-nets Alon, Noga Discrete & Computational Geometry, Vol. 47, Issue 2 https://doi.org/10.1007/s00454-010-9323-7	journal	January 2011
Approximation algorithms for hitting objects with straight lines Hassin, Refael; Megiddo, Nimrod Discrete Applied Mathematics, Vol. 30, Issue 1 https://doi.org/10.1016/0166-218X(91)90011-K	journal	January 1991
On the complexity of locating linear facilities in the plane Megiddo, Nimrod; Tamir, Arie Operations Research Letters, Vol. 1, Issue 5 https://doi.org/10.1016/0167-6377(82)90039-6	journal	November 1982
Art Gallery and Illumination Problems Urrutia, Jorge Handbook of Computational Geometry https://doi.org/10.1016/B978-044482537-7/50023-1	book	January 2000
Fixed-parameter tractability and lower bounds for stabbing problems Giannopoulos, Panos; Knauer, Christian; Rote, Günter Computational Geometry, Vol. 46, Issue 7 https://doi.org/10.1016/j.comgeo.2011.06.005	journal	October 2013
On the approximability of covering points by lines and related problems Dumitrescu, Adrian; Jiang, Minghui Computational Geometry, Vol. 48, Issue 9 https://doi.org/10.1016/j.comgeo.2015.06.006	journal	October 2015
Hitting sets when the VC-dimension is small Even, Guy; Rawitz, Dror; Shahar, Shimon (Moni) Information Processing Letters, Vol. 95, Issue 2 https://doi.org/10.1016/j.ipl.2005.03.010	journal	July 2005
Tight lower bounds for the size of epsilon-nets Pach, János; Tardos, Gábor Journal of the American Mathematical Society, Vol. 26, Issue 3 https://doi.org/10.1090/S0894-0347-2012-00759-0	journal	November 2012
Small-Size $\eps$-Nets for Axis-Parallel Rectangles and Boxes Aronov, Boris; Ezra, Esther; Sharir, Micha SIAM Journal on Computing, Vol. 39, Issue 7 https://doi.org/10.1137/090762968	journal	January 2010
Point Line Cover: The Easy Kernel is Essentially Tight Kratsch, Stefan; Philip, Geevarghese; Ray, Saurabh Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms https://doi.org/10.1137/1.9781611973402.116	conference	January 2014
Approximation Algorithms for Rectangle Stabbing and Interval Stabbing Problems Kovaleva, Sofia; Spieksma, Frits C. R. SIAM Journal on Discrete Mathematics, Vol. 20, Issue 3 https://doi.org/10.1137/S089548010444273X	journal	January 2006
Algorithms for capacitated rectangle stabbing and lot sizing with joint set-up costs Even, Guy; Levi, Retsef; Rawitz, Dror ACM Transactions on Algorithms, Vol. 4, Issue 3 https://doi.org/10.1145/1367064.1367074	journal	June 2008
An optimal algorithm for finding segments intersections Balaban, Ivan J. Proceedings of the eleventh annual symposium on Computational geometry - SCG '95 https://doi.org/10.1145/220279.220302	conference	January 1995
Approximation of k -set cover by semi-local optimization Duh, Rong-chii; Fürer, Martin Proceedings of the twenty-ninth annual ACM symposium on Theory of computing - STOC '97 https://doi.org/10.1145/258533.258599	conference	January 1997
Analytical approach to parallel repetition Dinur, Irit; Steurer, David Proceedings of the 46th Annual ACM Symposium on Theory of Computing - STOC '14 https://doi.org/10.1145/2591796.2591884	conference	January 2014
A Greedy Heuristic for the Set-Covering Problem Chvatal, V. Mathematics of Operations Research, Vol. 4, Issue 3 https://doi.org/10.1287/moor.4.3.233	journal	August 1979

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Graphs with equal domination and covering numbers Lingas, Andrzej; Miotk, Mateusz; Topp, Jerzy Journal of Combinatorial Optimization, Vol. 39, Issue 1 https://doi.org/10.1007/s10878-019-00454-6	journal	October 2019
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