Abstract
We design approximation algorithms for several NP-hard combinatorial problems achieving ratios that cannot be achieved in polynomial time (unless a very unlikely complexity conjecture is confirmed) with worst-case complexity much lower (though super-polynomial) than that of an exact computation. We study in particular max independent set, min vertex cover and min set cover and then extend our results to max clique, max bipartite subgraph and max set packing.
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
Woeginger, G.J.: Exact algorithms for NP-hard problems: a survey. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization - Eureka, You Shrink! LNCS, vol. 2570, pp. 185–207. Springer, Heidelberg (2003)
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and approximation. In: Combinatorial optimization problems and their approximability properties. Springer, Berlin (1999)
Hochbaum, D.S. (ed.): Approximation algorithms for NP-hard problems. PWS, Boston (1997)
Vazirani, V.: Approximation algorithms. Springer, Berlin (2001)
Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and intractability of approximation problems. J. Assoc. Comput. Mach. 45, 501–555 (1998)
Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. In: Proc. STOC 2006, pp. 681–690 (2006)
Björklund, A., Husfeldt, T.: Inclusion-exclusion algorithms for counting set partitions. In: Proc. FOCS 2006, pp. 575–582 (2006)
Cygan, M., Kowalik, L., Pilipczuk, M., Wykurz, M.: Exponential-time approximation of hard problems. arXiv:0810.4934v1 (2008), http://arxiv.org/abs/0810.4934v1
Cai, L., Huang, X.: Fixed-parameter approximation: conceptual framework and approximability results. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 96–108. Springer, Heidelberg (2006)
Chen, Y., Grohe, M., Grüber, M.: On parameterized approximability. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 109–120. Springer, Heidelberg (2006)
Downey, R.G., Fellows, M.R., McCartin, C.: Parameterized approximation problems. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 121–129. Springer, Heidelberg (2006)
Vassilevska, V., Williams, R., Woo, S.: Confronting hardness using a hybrid approach. In: Proc. Symposium on Discrete Algorithms, SODA 2006, pp. 1–10 (2006)
Bourgeois, N., Escoffier, B., Paschos, V.T.: Efficient approximation by “low-complexity” exponential algorithms. Cahier du LAMSADE 271, LAMSADE, Université Paris-Dauphine (2007), http://www.lamsade.dauphine.fr/cahiers/PDF/cahierLamsade271.pdf
Bourgeois, N., Escoffier, B., Paschos, V.T.: Efficient approximation of min set cover by “low-complexity” exponential algorithms. Cahier du LAMSADE 278, LAMSADE, Université Paris-Dauphine (2008), http://www.lamsade.dauphine.fr/cahiers/PDF/cahierLamsade278.pdf
Robson, J.M.: Finding a maximum independent set in time O(2n/4). Technical Report 1251-01, LaBRI, Université de Bordeaux I (2001)
Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2 − ε. In: Proc. Annual Conference on Computational Complexity, CCC 2003, pp. 379–386 (2003)
Chen, J., Kanj, I., Jia, W.: Vertex cover: further observations and further improvements. J. Algorithms 41, 280–301 (2001)
Nemhauser, G.L., Trotter, L.E.: Vertex packings: structural properties and algorithms. Math. Programming 8, 232–248 (1975)
Berman, P., Fujito, T.: On the approximation properties of independent set problem in degree 3 graphs. In: Sack, J.-R., Akl, S.G., Dehne, F., Santoro, N. (eds.) WADS 1995. LNCS, vol. 955, pp. 449–460. Springer, Heidelberg (1995)
Duh, R., Fürer, M.: Approximation of k-set cover by semi-local optimization. In: Proc. STOC 1997, pp. 256–265 (1997)
van Rooij, J., Bodlaender, H.: Design by measure and conquer, a faster exact algorithm for dominating set. In: Albers, S., Weil, P. (eds.) Proc. International Symposium on Theoretical Aspects of Computer Science, STACS 2008, pp. 657–668 (2008)
Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion-exclusion. SIAM J. Comput. (To appear in the special issue dedicated to selected papers from FOCS 2006)
Simon, H.U.: On approximate solutions for combinatorial optimization problems. SIAM J. Disc. Math. 3, 294–310 (1990)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bourgeois, N., Escoffier, B., Paschos, V.T. (2009). Efficient Approximation of Combinatorial Problems by Moderately Exponential Algorithms. In: Dehne, F., Gavrilova, M., Sack, JR., Tóth , C.D. (eds) Algorithms and Data Structures. WADS 2009. Lecture Notes in Computer Science, vol 5664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03367-4_44
Download citation
DOI: https://doi.org/10.1007/978-3-642-03367-4_44
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03366-7
Online ISBN: 978-3-642-03367-4
eBook Packages: Computer ScienceComputer Science (R0)