Abstract
In the unweighted set-cover problem we are given a set of elements E={ e 1,e 2, ...,e n } and a collection \(\cal F\) of subsets of E. The problem is to compute a sub-collection SOL ⊆\(\cal F\) such that \(\bigcup_{S_j\in SOL}S_j=E\) and its size |SOL| is minimized. When |S|≤k for all \(S\in\cal F\) we obtain the unweighted k-set cover problem. It is well known that the greedy algorithm is an H k -approximation algorithm for the unweighted k-set cover, where \(H_k=\sum_{i=1}^k {1 \over i}\) is the k-th harmonic number, and that this bound on the approximation ratio of the greedy algorithm, is tight for all constant values of k. Since the set cover problem is a fundamental problem, there is an ongoing research effort to improve this approximation ratio using modifications of the greedy algorithm. The previous best improvement of the greedy algorithm is an \(\left( H_k-{1\over 2}\right)\)-approximation algorithm. In this paper we present a new \(\left( H_k-{196\over 390}\right)\)-approximation algorithm for k ≥4 that improves the previous best approximation ratio for all values of k≥4 . Our algorithm is based on combining local search during various stages of the greedy algorithm.
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
Bar-Yehuda, R., Even, S.: A linear time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms 2, 198–203 (1981)
Chvátal, V.: A greedy heuristic for the set-covering problem. Mathematics of Operations Research 4, 233–235 (1979)
Crescenzi, P., Kann, V.: A compendium of NP optimization problems (1995), http://www.nada.kth.se/theory/problemlist.html
Duh, R., Fürer, M.: Approximation of k-set cover by semi local optimization. In: Proc. STOC 1997, pp. 256–264 (1997)
Feige, U.: A threshold of ln n for approximating set cover. Journal of the ACM 45, 634–652 (1998)
Fujito, T., Okumura, T.: A modified greedy algorithm for the set cover problem with weights 1 and 2. In: Eades, P., Takaoka, T. (eds.) ISAAC 2001. LNCS, vol. 2223, pp. 670–681. Springer, Heidelberg (2001)
Goldschmidt, O., Hochbaum, D.S., Yu, G.: A modified greedy heuristic for the set covering problem with improved worst case bound. Information Processing Letters 48, 305–310 (1993)
Halldórsson, M.M.: Approximating k set cover and complementary graph coloring. In: Cunningham, W.H., Queyranne, M., McCormick, S.T. (eds.) IPCO 1996. LNCS, vol. 1084, pp. 118–131. Springer, Heidelberg (1996)
Hassin, R., Levin, A.: A better-than-greedy approximation algorithm for the minimum set cover problem. SIAM J. Computing 35, 189–200 (2006)
Hochbaum, D.S.: Approximation algorithms for the weighted set covering and node cover problems. SIAM Journal on Computing 11, 555–556 (1982)
Hurkens, C.A.J., Schrijver, A.: On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems. SIAM Journal on Discrete Mathematics 2, 68–72 (1989)
Johnson, D.S.: Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences 9, 256–278 (1974)
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of computer computations, pp. 85–103. Plenum Press, New-York (1972)
Khanna, S., Motwani, R., Sudan, M., Vazirani, U.V.: On syntactic versus computational views of approximability. SIAM Journal on Computing 28, 164–191 (1998)
Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Mathematics 13, 383–390 (1975)
Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation and complexity classes. Journal of Computer System Sciences 43, 425–440 (1991)
Paschos, V.T.: A survey of approximately optimal solutions to some covering and packing problems. ACM Computing Surveys 29, 171–209 (1997)
Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and sub-constant error-probability PCP characterization of NP. In: Proc. STOC 1997, pp. 475–484 (1997)
Slavík, P.: A tight analysis of the greedy algorithm for set cover. Journal of Algorithms 25, 237–254 (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Levin, A. (2007). Approximating the Unweighted k-Set Cover Problem: Greedy Meets Local Search. In: Erlebach, T., Kaklamanis, C. (eds) Approximation and Online Algorithms. WAOA 2006. Lecture Notes in Computer Science, vol 4368. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11970125_23
Download citation
DOI: https://doi.org/10.1007/11970125_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69513-4
Online ISBN: 978-3-540-69514-1
eBook Packages: Computer ScienceComputer Science (R0)