Learnability and the Vapnik-Chervonenkis dimension
Journal of the ACM (JACM)
Almost tight bounds for &egr;-nets
Discrete & Computational Geometry
A parallel approximation algorithm for positive linear programming
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Fast approximation algorithms for fractional packing and covering problems
Mathematics of Operations Research
Randomized rounding without solving the linear program
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
Lectures on Discrete Geometry
Sequential and Parallel Algorithms for Mixed Packing and Covering
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Epsilon nets and union complexity
Proceedings of the twenty-fifth annual symposium on Computational geometry
PTAS for geometric hitting set problems via local search
Proceedings of the twenty-fifth annual symposium on Computational geometry
Near-linear approximation algorithms for geometric hitting sets
Proceedings of the twenty-fifth annual symposium on Computational geometry
Approximation algorithms for maximum independent set of pseudo-disks
Proceedings of the twenty-fifth annual symposium on Computational geometry
On the set multi-cover problem in geometric settings
Proceedings of the twenty-fifth annual symposium on Computational geometry
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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 ε-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.