Abstract
Let ℱ∪{U} be a collection of convex sets in ℝd such that ℱ covers U. We prove that if the elements of ℱ and U have comparable size, then a small subset of ℱ suffices to cover most of the volume of U; we analyze how small this subset can be depending on the geometry of the elements of ℱ and show that smooth convex sets and axis parallel squares behave differently. We obtain similar results for surface-to-surface visibility amongst balls in three dimensions for a notion of volume related to form factor. For each of these situations, we give an algorithm that takes ℱ and U as input and computes in time O(|ℱ|*|ℋ ε |) either a point in U not covered by ℱ or a subset ℋ ε covering U up to a measure ε, with |ℋ ε | meeting our combinatorial bounds.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Amenta, N.: Helly theorems and generalized linear programming. PhD thesis, U.C. Berkeley (1993)
Amenta, N.: Helly-type theorems and generalized linear programming. Discrete Comput. Geom. 12 (1994)
Barany, I., Katchalski, M., Pach, J.:: Quantitative Helly-type theorems. Proc. Am. Math. Soc. 86(1), 109–114 (1982)
Barany, I., Katchalski, M., Pach, J.: Helly’s theorem with volumes. Am. Math. Mon. 91(6), 362–365 (1984)
Bronnimann, H., Goodrich, M.T.: Almost optimal set covers in finite VC-dimension. Discrete Comput. Geom. 14, 463–479 (1995)
Clarkson, K.L., Varadarajan, K.: Improved approximation algorithms for geometric set cover. Discrete Comput. Geom. 37, 43–58 (2007)
Cohen, M.F., Wallace, J.R.: Radiosity and Realistic Image Synthesis. Academic Press, New York (1993)
Danzer, L., Grünbaum, B., Klee, V.: Helly’s theorem and its relatives. In: Klee, V. (ed.) Convexity, Proceedings of Symposia in Pure Mathematics, pp. 101–180. Amer. Math. Soc., Reading (1963)
Eckhoff, J.: Helly, Radon and Carathéodory type theorems. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, pp. 389–448. North Holland, Amsterdam (1993)
Feige, U.: A threshold of lnn for approximating set cover. J. ACM 45(4), 634–652 (1998)
Gärtner, B., Welzl, E.: Linear programming—randomization and abstract frameworks. In: 13th Annual Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 1046, pp. 669–687. Springer, Berlin (1996)
Karp, R.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, Proc. Sympos. IBM Thomas J. Watson Res. Center, pp. 85–103 (1972)
Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. J. ACM 41(5), 960–981 (1994)
Golin, M.: How many maxima can there be? Comput. Geom. Theory Appl. 2, 335–353 (1993)
Pach, J., Sharir, M.: Combinatorial Geometry with Algorithmic Applications, The Alcala Lectures, Alcala, Spain, August 31–September 5, 2006
Pellegrini, M.: Monte Carlo approximation of form factors with error bounded a priori. In: Symposium on Computational Geometry, pp. 287–296 (1995)
Arya, S., Malamatos, T., Mount, D.M.: The effect of corners on the complexity of approximate range searching. In: Symposium on Computational Geometry, pp. 11–20 (2006)
Santalo, L.A.: Integral Geometry and Geometric Probability, 2nd edn. Cambridge University Press, New York (2004)
Sharir, M., Welzl, E.: A combinatorial bound for linear programming and related problems. In: STACS ’92: Proceedings of the 9th Annual Symposium on Theoretical Aspects of Computer Science, pp. 569–579. Springer, Berlin (1992)
Sillion, F., Puech, C.: Radiosity and Global Illumination. Morgan Kaufmann, San Mateo (1994)
Wenger, R.: Helly-type theorems and geometric transversals. In: Goodman, J.E., O’Raurke, J., (eds.) Handbook of Discrete & Computational Geometry, 2nd edn., pp. 73–96. CRC Press, Boca Raton (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors acknowledge support from the French–Korean Science and Technology amicable relationship program (STAR) 11844QJ.
Rights and permissions
About this article
Cite this article
Demouth, J., Devillers, O., Glisse, M. et al. Helly-Type Theorems for Approximate Covering. Discrete Comput Geom 42, 379–398 (2009). https://doi.org/10.1007/s00454-009-9167-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-009-9167-1