Abstract
We consider the frugal coverage problem, an interesting variation of set cover defined as follows. Instances of the problem consist of a universe of elements and a collection of sets over these elements; the objective is to compute a subcollection of sets so that the number of elements it covers plus the number of sets not chosen is maximized. The problem was introduced and studied by Huang and Svitkina [7] due to its connections to the donation center location problem. We prove that the greedy algorithm has approximation ratio at least 0.782, improving a previous bound of 0.731 in [7]. We also present a further improvement that is obtained by adding a simple corrective phase at the end of the execution of the greedy algorithm. The approximation ratio achieved in this way is at least 0.806. Our analysis is based on the use of linear programs which capture the behavior of the algorithms in worst-case examples. The obtained bounds are proved to be tight.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Athanassopoulos, S., Caragiannis, I., Kaklamanis, C.: Analysis of approximation algorithms for k-set cover using factor-revealing linear programs. Theory of Computing Systems 45(3), 555–576 (2009)
Athanassopoulos, S., Caragiannis, I., Kaklamanis, C., Kyropoulou, M.: An improved approximation bound for spanning star forest and color saving. In: Královič, R., Niwiński, D. (eds.) MFCS 2009. LNCS, vol. 5734, pp. 90–101. Springer, Heidelberg (2009)
Caragiannis, I.: Wavelength management in WDM rings to maximize the number of connections. SIAM Journal on Discrete Mathematics 23(2), 959–978 (2009)
Duh, R., Fürer, M.: Approximation of k-set cover by semi local optimization. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing (STOC), pp. 256–264 (1997)
Feige, U.: A threshold of ln n for approximating set cover. Journal of the ACM 45(4), 634–652 (1998)
Feige, U., Jozeph, S.: Oblivious algorithms for the maximum directed cut problem. arXiv: 1010.0406 (2010)
Huang, C.-C., Svitkina, Z.: Donation center location problem. In: Proceedings of the 29th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), pp. 227–238 (2009)
Jain, K., Mahdian, M., Markakis, E., Saberi, A., Vazirani, V.V.: Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. Journal of the ACM 50(6), 795–824 (2003)
Johnson, D.S.: Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences 9, 256–278 (1974)
Levin, A.: Approximating the unweighted k-set cover problem: greedy meets local search. SIAM Journal on Discrete Mathematics 23(1), 251–264 (2008)
Levin, A., Yovel, U.: Uniform unweighted set cover: the power of non-oblivious local search. Theoretical Computer Science 412(12-14), 1033–1053 (2011)
Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and sub-constant error-probability PCP characterization of NP. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing (STOC), pp. 475–484 (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Caragiannis, I., Kaklamanis, C., Kyropoulou, M. (2011). Tight Approximation Bounds for Greedy Frugal Coverage Algorithms. In: Atallah, M., Li, XY., Zhu, B. (eds) Frontiers in Algorithmics and Algorithmic Aspects in Information and Management. Lecture Notes in Computer Science, vol 6681. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21204-8_22
Download citation
DOI: https://doi.org/10.1007/978-3-642-21204-8_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21203-1
Online ISBN: 978-3-642-21204-8
eBook Packages: Computer ScienceComputer Science (R0)