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# Applications of Incidence Bounds in Point Covering Problems

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LIPIcs.SoCG.2016.60.pdf
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## Cite As

Peyman Afshani, Edvin Berglin, Ingo van Duijn, and Jesper Sindahl Nielsen. Applications of Incidence Bounds in Point Covering Problems. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 60:1-60:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.SoCG.2016.60

## Abstract

In the Line Cover problem a set of n points is given and the task is to cover the points using either the minimum number of lines or at most k lines. In Curve Cover, a generalization of Line Cover, the task is to cover the points using curves with d degrees of freedom. Another generalization is the Hyperplane Cover problem where points in d-dimensional space are to be covered by hyperplanes. All these problems have kernels of polynomial size, where the parameter is the minimum number of lines, curves, or hyperplanes needed. First we give a non-parameterized algorithm for both problems in O*(2^n) (where the O*(.) notation hides polynomial factors of n) time and polynomial space, beating a previous exponential-space result. Combining this with incidence bounds similar to the famous Szemeredi-Trotter bound, we present a Curve Cover algorithm with running time O*((Ck/log k)^((d-1)k)), where C is some constant. Our result improves the previous best times O*((k/1.35)^k) for Line Cover (where d=2), O*(k^(dk)) for general Curve Cover, as well as a few other bounds for covering points by parabolas or conics. We also present an algorithm for Hyperplane Cover in R^3 with running time O*((Ck^2/log^(1/5) k)^k), improving on the previous time of O*((k^2/1.3)^k).
##### Keywords
• Point Cover
• Incidence Bounds
• Inclusion Exclusion
• Exponential Algorithm

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## References

1. Peyman Afshani, Edvin Berglin, Ingo van Duijn, and Jesper Sindahl Nielsen. Applications of incidence bounds in point covering problems. arXiv:1603.07282, 2016.
2. Pankaj K Agarwal and Boris Aronov. Counting facets and incidences. Discrete &Computational Geometry, 7(1):359-369, 1992.
3. Andreas Björklund, Thore Husfeldt, and Mikko Koivisto. Set partitioning via inclusion-exclusion. SIAM Journal on Computing, 39(2):546-563, 2009.
4. Cheng Cao. Study on two optimization problems: Line cover and maximum genus embedding. Master’s thesis, Texas A&M University, 2012.
5. Herbert Edelsbrunner. Algorithms in Combinatorial Geometry. Springer Publishing Company, Incorporated, 1st edition, 2012.
6. Herbert Edelsbrunner, Leonidas Guibas, and Micha Sharir. The complexity of many cells in arrangements of planes and related problems. Discrete &Computational Geometry, 5(1):197-216, 1990.
7. György Elekes and Csaba D Tóth. Incidences of not-too-degenerate hyperplanes. In Proceedings of the twenty-first annual symposium on Computational geometry, pages 16-21. ACM, 2005.
8. Vladimir Estivill-Castro, Apichat Heednacram, and Francis Suraweera. FPT-algorithms for minimum-bends tours. International Journal of Computational Geometry &Applications, 21(02):189-213, 2011.
9. Jacob Fox, János Pach, Adam Sheffer, Andrew Suk, and Joshua Zahl. A semi-algebraic version of Zarankiewicz’s problem. arXiv preprint arXiv:1407.5705, 2014.
10. Magdalene Grantson and Christos Levcopoulos. Covering a set of points with a minimum number of lines. Springer, 2006.
11. Ben Joseph Green and Terence Tao. On sets defining few ordinary lines. Discrete & Computational Geometry, 50(2):409-468, 2013.
12. Leonidas J Guibas, Mark H Overmars, and Jean-Marc Robert. The exact fitting problem in higher dimensions. Computational geometry, 6(4):215-230, 1996.
13. LJ Guibas, Mark Overmars, and Jean-Marc Robert. The exact fitting problem for points. In Proc. 3rd Canadian Conference on Computational Geometry, pages 171-174, 1991.
14. Stefan Kratsch, Geevarghese Philip, and Saurabh Ray. Point line cover: The easy kernel is essentially tight. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1596-1606. SIAM, 2014.
15. VS Anil Kumar, Sunil Arya, and Hariharan Ramesh. Hardness of set cover with intersection 1. In Automata, Languages and Programming, pages 624-635. Springer, 2000.
16. Stefan Langerman and Pat Morin. Covering things with things. Discrete &Computational Geometry, 33(4):717-729, 2005.
17. Nimrod Megiddo and Arie Tamir. On the complexity of locating linear facilities in the plane. Operations research letters, 1(5):194-197, 1982.
18. János Pach and Micha Sharir. On the number of incidences between points and curves. Combinatorics, Probability and Computing, 7(01):121-127, 1998.
19. József Solymosi and Terence Tao. An incidence theorem in higher dimensions. Discrete &Computational Geometry, 48(2):255-280, 2012.
20. Endre Szemerédi and William T Trotter Jr. Extremal problems in discrete geometry. Combinatorica, 3(3-4):381-392, 1983.
21. Praveen Tiwari. On covering points with conics and strips in the plane. Master’s thesis, Texas A&M University, 2012.
22. Jianxin Wang, Wenjun Li, and Jianer Chen. A parameterized algorithm for the hyperplane-cover problem. Theoretical Computer Science, 411(44):4005-4009, 2010.
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