Hitting sets when the VC-dimension is small

https://doi.org/10.1016/j.ipl.2005.03.010Get rights and content

Abstract

We present an approximation algorithm for the hitting set problem when the VC-dimension of the set system is small. Our algorithm uses a linear programming relaxation to compute a probability measure for which ɛ-nets are always hitting sets (see Corollary 15.6 in Pach and Agarwal [Combinatorial Geometry, J. Wiley, New York, 1995]). The comparable algorithm of Brönnimann and Goodrich [Almost optimal set covers in finite VC-dimension, Discrete Comput. Geom. 14 (1995) 463] computes such a probability measure by an iterative reweighting technique. The running time of our algorithm is comparable with theirs, and the approximation ratio is smaller by a constant factor. We also show how our algorithm can be parallelized and extended to the minimum cost hitting set problem.

References (12)

  • A. Blumer et al.

    Learnability and the Vapnik–Chervonenkis dimension

    J. ACM

    (1989)
  • H. Brönnimann et al.

    Almost optimal set covers in finite VC-dimension

    Discrete Comput. Geom.

    (1995)
  • D. Haussler et al.

    ɛ-nets and simplex range queries

    Discrete Comput. Geom.

    (1987)
  • J. Komlós et al.

    Almost tight bounds for ɛ-nets

    Discrete Comput. Geom.

    (1992)
  • M. Luby, N. Nisan, A parallel approximation algorithm for positive linear programming, in: ACM Symp. on Theory of...
  • S. Mahajan et al.

    Solving some discrepancy problems in NC

    Algorithmica

    (2001)
There are more references available in the full text version of this article.

Cited by (115)

  • Distributed domination on sparse graph classes

    2023, European Journal of Combinatorics
  • Geometric covering via extraction Theorem

    2024, Leibniz International Proceedings in Informatics, LIPIcs
  • Covering Rectilinear Polygons with Area-Weighted Rectangles

    2024, Proceedings of the Workshop on Algorithm Engineering and Experiments
View all citing articles on Scopus
View full text