# Kernelization of the subset general position problem in geometry

Jean Daniel Boissonnat, Kunal Dutta, Arijit Ghosh, Sudeshna Kolay

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### Abstract

In this paper, we consider variants of the Geometric Subset General Position problem. In defining this problem, a geometric subsystem is specified, like a subsystem of lines, hyperplanes or spheres. The input of the problem is a set of n points in Rd and a positive integer k. The objective is to find a subset of at least k input points such that this subset is in general position with respect to the specified subsystem. For example, a set of points is in general position with respect to a subsystem of hyperplanes in Rd if no d + 1 points lie on the same hyperplane. In this paper, we study the Hyperplane Subset General Position problem under two parameterizations. When parameterized by k then we exhibit a polynomial kernelization for the problem. When parameterized by h = n - k, or the dual parameter, then we exhibit polynomial kernels which are also tight, under standard complexity theoretic assumptions. We can also exhibit similar kernelization results for d-Polynomial Subset General Position, where a vector space of polynomials of degree at most d are specified as the underlying subsystem such that the size of the basis for this vector space is b. The objective is to find a set of at least k input points, or in the dual delete at most h = n-k points, such that no b+1 points lie on the same polynomial. Notice that this is a generalization of many well-studied geometric variants of the Set Cover problem, such as Circle Subset General Position. We also study general projective variants of these problems. These problems are also related to other geometric problems like Subset Delaunay Triangulation problem.

Original language English 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017 Kim G. Larsen, Hans L. Bodlaender, Jean-Francois Raskin Dagstuhl Schloss Dagstuhl - Leibniz-Zentrum für Informatik 978-3-95977-046-0 https://doi.org/10.4230/LIPIcs.MFCS.2017.25 Published - 1 Nov 2017 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017) - Aalborg, DenmarkDuration: 21 Aug 2017 → 25 Aug 2017Conference number: 42http://mfcs2017.cs.aau.dk/

### Publication series

Name Leibniz International Proceedings in Informatics (LIPIcs) 83

### Conference

Conference 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017) MFCS 2017 Denmark Aalborg 21/08/17 → 25/08/17 http://mfcs2017.cs.aau.dk/

### Keywords

• Bounded degree polynomials
• Hyperplanes
• Incidence Geometry
• Kernel Lower bounds

• ## Cite this

Boissonnat, J. D., Dutta, K., Ghosh, A., & Kolay, S. (2017). Kernelization of the subset general position problem in geometry. In K. G. Larsen, H. L. Bodlaender, & J-F. Raskin (Eds.), 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017 (Leibniz International Proceedings in Informatics (LIPIcs); Vol. 83). Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.MFCS.2017.25