Years and Authors of Summarized Original Work
1974–1979; Chvátal, Johnson, Lovász, Stein
Problem Definition
Given a collection \(\mathcal{S}\) of sets over a universe U, a set cover \(C \subseteq \mathcal{S}\) is a subcollection of the sets whose union is U. The set-cover problem is, given \(\mathcal{S}\), to find a minimum-cardinality set cover. In the weighted set-cover problem, for each set \(s \in \mathcal{S}\), a weight w s ≥ 0 is also specified, and the goal is to find a set-cover C of minimum total weight \(\sum \limits _{S\in C}w_{S}\).
Weighted set cover is a special case of minimizing a linear function subject to a submodular constraint, defined as follows. Given a collection \(\mathcal{S}\) of objects, for each object s a nonnegative weight w s , and a nondecreasing submodular function \(f : 2^{\mathcal{S}}\rightarrow\mathbb{R}\), the goal is to find a subcollection \(C \subseteq \mathcal{S}\)such that \(f\left (C\right ) = f\left (\mathcal{S}\right )\) minimizing \(\sum...
Recommended Reading
Brönnimann H, Goodrich MT (1995) Almost optimal set covers in finite VC-dimension. Discret Comput Geom 14(4):463–479
Chvátal V (1979) A greedy heuristic for the set-covering problem. Math Oper Res 4(3):  233–235
Feige U (1998) A threshold of ln n for approximating set cover. J ACM 45(4):634–652
Gonzalez TF (2007) Handbook of approximation algorithms and metaheuristics. Chapman & Hall/CRC computer & information science series. Chapman & Hall/CRC, Boca Raton
Johnson DS (1974) Approximation algorithms for combinatorial problems. J Comput Syst Sci 9:256–278
Khuller S, Moss A, Naor J (1999) The budgeted maximum coverage problem. Inform Process Lett 70(1):39–45
Kolliopoulos SG, Young NE (2001) Tight approximation results for general covering integer programs. In: Proceedings of the forty-second annual IEEE symposium on foundations of computer science, Las Vegas, pp 522–528
Lovász L (1975) On the ratio of optimal integral and fractional covers. Discret Math 13:383–390
Lund C, Yannakakis M (1994) On the hardness of approximating minimization problems. J ACM 41(5):960–981
Nemhauser GL, Wolsey LA (1988) Integer and combinatorial optimization. Wiley, New York
Slavik P (1997) A tight analysis of the greedy algorithm for set cover. J Algorithms 25(2): 237–254
Srinivasan A (1995) Improved approximations of packing and covering problems. In: Proceedings of the twenty-seventh annual ACM symposium on theory of computing, Heraklion, pp 268–276
Stein SK (1974) Two combinatorial covering theorems. J Comb Theor A 16:391–397
van Santen JPH, Buchsbaum AL (1997) Methods for optimal text selection. In: Proceedings of the European conference on speech communication and technology, Rhodos, vol 2, pp 553–556
Vazirani VV (2001) Approximation algorithms. Springer, Berlin/Heidelberg
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this entry
Cite this entry
Young, N. (2014). Greedy Set-Cover Algorithms. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-3-642-27848-8_175-2
Download citation
DOI: https://doi.org/10.1007/978-3-642-27848-8_175-2
Received:
Accepted:
Published:
Publisher Name: Springer, Boston, MA
Online ISBN: 978-3-642-27848-8
eBook Packages: Springer Reference Computer SciencesReference Module Computer Science and Engineering