Approximating maximum independent set in k-clique-free graphs
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.  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
Key wordsGraph Maximum Independent Set k-Clique Algorithm Complexity Approximation
Unable to display preview. Download preview PDF.
- 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.Google Scholar
- P. Berman and M. Fürer, Approximating maximum independent sets by excluding subgraphs, Proc. Fifth ACM-SIAM Symp. on Discrete Algorithms, 1994, 365–371.Google Scholar
- J. A. Bondy and U. S. R. Murty, Graph Theory with Applications (Macmillan, London and Elsevier, New York, 1976).Google Scholar
- Y. Caro, New Results on the Independence Number, Technical Report, Tel-Aviv University, 1979.Google Scholar
- 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.Google Scholar
- J. Håstad, Clique is hard to approximate within n 1−e, 37th Annual Symposium on Foundations of Computer Science, 1996, 627–636.Google Scholar
- M. M. Halldórsson and J. Radhakrishnan, Improved approximations of Independent Sets in Bounded-Degree Graphs, SWAT'94, LNCS 824 (1994) 195–206.Google Scholar
- S. Khanna, R. Motwani, M. Sudan and U. Vazirani, On syntactic versus computational views of approximability, Proc. 35th IEEE FoCS, 1994, 819–830.Google Scholar
- V. K. Wei, A Lower Bound on the Stability Number of a Simple Graph, Technical memorandum, TM 81-11217-9, Bell laboratories, 1981.Google Scholar