Discrete & Computational Geometry

, Volume 14, Issue 4, pp 463–479

Almost optimal set covers in finite VC-dimension

Authors

  • H. Brönnimann
    • Department of Computer SciencePrinceton University
  • M. T. Goodrich
    • Department of Computer ScienceJohns Hopkins University
Article

DOI: 10.1007/BF02570718

Cite this article as:
Brönnimann, H. & Goodrich, M.T. Discrete & Computational Geometry (1995) 14: 463. doi:10.1007/BF02570718

Abstract

We give a deterministic polynomial-time method for finding a set cover in a set system (X, ℛ) of dual VC-dimensiond such that the size of our cover is at most a factor ofO(d log(dc)) from the optimal size,c. For constant VC-dimensional set systems, which are common in computational geometry, our method gives anO(logc) approximation factor. This improves the previous Θ(log⋎X⋎) bound of the greedy method and challenges recent complexity-theoretic lower bounds for set covers (which do not make any assumptions about the VC-dimension). We give several applications of our method to computational geometry, and we show that in some cases, such as those arising in three-dimensional polytope approximation and two-dimensional disk covering, we can quickly findO(c)-sized covers.

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Copyright information

© Springer-Verlag New York Inc. 1995