WADS 2011: Algorithms and Data Structures pp 49-60

# Convex Transversals

• Esther M. Arkin
• Claudia Dieckmann
• Christian Knauer
• Joseph S. B. Mitchell
• Valentin Polishchuk
• Lena Schlipf
• Shang Yang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6844)

## Abstract

We answer the question initially posed by Arik Tamir at the Fourth NYU Computational Geometry Day (March, 1987): “Given a collection of compact sets, can one decide in polynomial time whether there exists a convex body whose boundary intersects every set in the collection?”

We prove that when the sets are segments in the plane, deciding existence of the convex stabber is NP-hard. The problem remains NP-hard if the sets are regular polygons. We also show that in 3D the stabbing problem is hard when the sets are balls. On the positive side, we give a polynomial-time algorithm to find a convex transversal of a maximum number of pairwise-disjoint segments in 2D if the vertices of the transversal are restricted to a given set of points. Our algorithm also finds a convex stabber of the maximum number of a set of convex pseudodisks in the plane.

The stabbing problem is related to “convexity” of point sets measured as the minimum distance by which the points must be shifted in order to arrive in convex position; we give a PTAS to find the minimum shift in 2D, and a 2-approximation in any dimension. We also consider stabbing with vertices of a regular polygon – a problem closely related to approximate symmetry detection.

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### References

1. 1.
Arkin, E.M., Mitchell, J.S.B., Polishchuk, V., Yang, S.: Convex transversals. In: Fall Workshop on Computational Geometry (2010)Google Scholar
2. 2.
Basu, S., Pollack, R., Roy, M.-F.: On computing a set of points meeting every cell defined by a family of polynomials on a variety. J. Complex. 13(1), 28–37 (1997)
3. 3.
Efrat, A., Itai, A., Katz, M.J.: Geometry helps in bottleneck matching and related problems. Algorithmica 31(1), 1–28 (2001)
4. 4.
Goodrich, M.T., Snoeyink, J.: Stabbing parallel segments with a convex polygon. Comput. Vision Graph. Image Process. 49(2), 152–170 (1990)
5. 5.
Iwanowski, S.: Testing approximate symmetry in the plane is NP-hard. Theor. Comput. Sci. 80(2), 227–262 (1991)
6. 6.
Kaplan, H., Rubin, N., Sharir, M.: Line transversals of convex polyhedra in ℝ3. In: SODA 2009, pp. 170–179 (2009)Google Scholar
7. 7.
Löffler, M., van Kreveld, M.J.: Largest and smallest convex hulls for imprecise points. Algorithmica 56(2), 235–269 (2010)
8. 8.
Tamir, A.: Problem 4-2 (New York University, Dept. of Statistics and Operations Research). Problems Presented at the Fourth NYU Computational Geometry Day (March 13, 1987)Google Scholar

© Springer-Verlag Berlin Heidelberg 2011

## Authors and Affiliations

• Esther M. Arkin
• 1
• Claudia Dieckmann
• 2
• Christian Knauer
• 3
• Joseph S. B. Mitchell
• 1
• Valentin Polishchuk
• 4
• Lena Schlipf
• 2
• Shang Yang
• 5
1. 1.Department of Applied Mathematics and StatisticsStony Brook UniversityUSA
2. 2.Institute of Computer ScienceFreie Universität BerlinGermany
3. 3.Institute of Computer ScienceUniversität BayreuthGermany
4. 4.Helsinki Institute for Information Technology, CS DeptUniversity of HelsinkiFinland
5. 5.Department of Computer ScienceStony Brook UniversityUSA