Abstract
In this paper, we consider generalizations of classical covering problems to handle hard capacities. In the hard capacitated set cover problem, additionally each set has a covering capacity which we are not allowed to exceed. In other words, after picking a set, we may cover at most a specified number of elements. Based on the classical results by Wolsey, an O(logn) approximation follows for this problem.
Chuzhoy and Naor [FOCS 2002], first studied the special case of unweighted vertex cover with hard capacities and developed an elegant 3 approximation for it based on rounding a natural LP relaxation. This was subsequently improved to a 2 approximation by Gandhi et al. [ICALP 2003]. These results are surprising in light of the fact that for weighted vertex cover with hard capacities, the problem is at least as hard as set cover to approximate. Hence this separates the unweighted problem from the weighted version.
The set cover hardness precludes the possibility of a constant factor approximation for the hard-capacitated vertex cover problem on weighted graphs. However, it was not known whether a better than logarithmic approximation is possible on unweighted multigraphs, i.e., graphs that may contain parallel edges. Neither the approach of Chuzhoy and Naor, nor the follow-up work of Gandhi et al. can handle the case of multigraphs. In fact, achieving a constant factor approximation for hard-capacitated vertex cover problem on unweighted multigraphs was posed as an open question in Chuzhoy and Naor’s work. In this paper, we resolve this question by providing the first constant factor approximation algorithm for the vertex cover problem with hard capacities on unweighted multigraphs. Previous works cannot handle hypergraphs which is analogous to consider set systems where elements belong to at most f sets. In this paper, we give an O(f) approximation algorithm for this problem. Further, we extend these works to consider partial covers.
Research supported by NSF CCF-0728839, NSF CCF-0937865 and a Google Research Award.
This is a preview of subscription content, log in via an institution to check access.
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-Ilan, J., Kortsarz, G., Peleg, D.: Generalized submodular cover problems and applications. Theor. Comput. Sci. 250, 179–200 (2001)
Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics 25, 27–45 (1985)
Bar-Yehuda, R., Flysher, G., Mestre, J., Rawitz, D.: Approximation of Partial Capacitated Vertex Cover. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 335–346. Springer, Heidelberg (2007)
Chuzhoy, J., Naor (Seffi)., J.: Covering problems with hard capacities. SIAM J. Comput. 36(2), 498–515 (2006)
Chuzhoy, J., Rabani, Y.: Approximating k-median with non-uniform capacities. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, pp. 952–958 (2005)
Demaine, E.D., Zadimoghaddam, M.: Scheduling to minimize power consumption using submodular functions. In: Proceedings of the 22nd ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2010, pp. 21–29 (2010)
Gandhi, R., Halperin, E., Khuller, S., Kortsarz, G., Srinivasan, A.: An improved approximation algorithm for vertex cover with hard capacities. J. Comput. Syst. Sci. 72, 16–33 (2006)
Gandhi, R., Khuller, S., Srinivasan, A.: Approximation algorithms for partial covering problems. J. Algorithms 53(1), 55–84 (2004)
Guha, S., Hassin, R., Khuller, S., Or, E.: Capacitated vertex covering. Journal of Algorithms 48(1), 257–270 (2003)
Hochbaum, D.S.: Approximation algorithms for the set covering and vertex cover problems. Siam Journal on Computing 11, 555–556 (1982)
Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9, 256–278 (1974)
Khuller, S., Li, J., Saha, B.: Energy efficient scheduling via partial shutdown. In: SODA, pp. 1360–1372 (2010)
Kolliopoulos, S.G.: Approximating covering integer programs with multiplicity constraints. Discrete Appl. Math. 129, 461–473 (2003)
Kolliopoulos, S.G., Young, N.E.: Tight approximation results for general covering integer programs. In: IEEE Symposium on Foundations of Computer Science, pp. 522–528 (2001)
Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Mathematics 13(4), 383–390 (1975)
Mahdian, M., Pal, M.: Universal facility location. In: Proc. of European Symposium of Algorithms 2003, pp. 409–421 (2003)
Mestre, J.: A Primal-Dual Approximation Algorithm for Partial Vertex Cover: Making Educated Guesses. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 182–191. Springer, Heidelberg (2005)
Wolsey, L.A.: An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica 2, 385–393 (1982)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Saha, B., Khuller, S. (2012). Set Cover Revisited: Hypergraph Cover with Hard Capacities. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31594-7_64
Download citation
DOI: https://doi.org/10.1007/978-3-642-31594-7_64
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31593-0
Online ISBN: 978-3-642-31594-7
eBook Packages: Computer ScienceComputer Science (R0)