Abstract
In this paper we study lower bounds and approximation algorithms for the independence number α(G) in k-clique-free graphs G. Ajtai et al. [1] showed that there exists an absolute constant c 1 such that for any k-clique-free graph G on n vertices and with average degree ¯d, α(G) ≥ c 1 log((log ¯d)/k)/d n
We improve this lower bound for α(G) as follows: Let G be a connected k-clique-free graph on n vertices with maximum degree δ(G) ≤ n − 2. Then α(G) ≥ n(¯d(k − 2)2 log(¯d(k − 2)2) − ¯d(k − 2)2 + 1)/(¯d(k − 2)2 − 1)2 for ¯d ≥ 2.
For graphs with moderate maximum degree Halldórsson and J. Radhakrishnan
For graphs with moderate to large values of δ Halldórsson and J. Radhakrishnan
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
M. Ajtai, P. Erdös, J. KomlⓈ and E. Szemerédi, On Turán's theorem for sparse graphs, Combinatorica 1 (4) (1981) 313–317.
S. Arora, C. Lund, R. Motwani, M. Sudan and M. Szegedy, Proof verification and hardness of approximation problems, Proc. 33rd IEEE FoCS, 1992, 14–23.
P. Berman and M. Fürer, Approximating maximum independent sets by excluding subgraphs, Proc. Fifth ACM-SIAM Symp. on Discrete Algorithms, 1994, 365–371.
R. B. Boppana and M. M. Halldórsson, Approximating maximum independent set by excluding subgraphs, BIT 32 (1992) 180–196.
J. A. Bondy and U. S. R. Murty, Graph Theory with Applications (Macmillan, London and Elsevier, New York, 1976).
Y. Caro, New Results on the Independence Number, Technical Report, Tel-Aviv University, 1979.
M. R. Garey and D. S. Johnson, Computers and Intractability, A Guide to the Theory of NP-Completeness, W. H. Freeman and Company, New York, 1979.
J. Håstad, Clique is hard to approximate within n 1−e, 37th Annual Symposium on Foundations of Computer Science, 1996, 627–636.
M. M. Halldórsson and J. Radhakrishnan, Improved approximations of Independent Sets in Bounded-Degree Graphs, SWAT'94, LNCS 824 (1994) 195–206.
M. M. Halldórsson and J. Radhakrishnan, Greed is Good: Approximating Independent Sets in Sparse and Bounded-Degree Graphs, Algorithmica 18 (1997) 145–163.
S. Khanna, R. Motwani, M. Sudan and U. Vazirani, On syntactic versus computational views of approximability, Proc. 35th IEEE FoCS, 1994, 819–830.
G. L. Nemhauser and L. E. Trotter Jr., Vertex Packings: Structural Properties and Algorithms, Mathematical Programming 8 (1975) 232–248.
J. B. Shearer, A Note on the Independence Number of Triangle-Free Graphs, Discrete Math. 46 (1983) 83–87.
J. B. Shearer, A Note on the Independence Number of Triangle-Free Graphs, II, J. Combin. Ser. B 53 (1991) 300–307.
J. B. Shearer, On the Independence Number of Sparse Graphs, Random Structures and Algorithms 5 (1995) 269–271.
P. Turán, On an extremal problem in graph theory (in Hungarian), Mat. Fiz. Lapok 48 (1941) 436–452.
V. K. Wei, A Lower Bound on the Stability Number of a Simple Graph, Technical memorandum, TM 81-11217-9, Bell laboratories, 1981.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Schiermeyer, I. (1998). Approximating maximum independent set in k-clique-free graphs. In: Jansen, K., Rolim, J. (eds) Approximation Algorithms for Combinatiorial Optimization. APPROX 1998. Lecture Notes in Computer Science, vol 1444. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0053972
Download citation
DOI: https://doi.org/10.1007/BFb0053972
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64736-2
Online ISBN: 978-3-540-69067-2
eBook Packages: Springer Book Archive