Advertisement

Approximating maximum independent set in k-clique-free graphs

  • Ingo Schiermeyer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1444)

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)nd(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 words

Graph Maximum Independent Set k-Clique Algorithm Complexity Approximation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Ajtai, P. Erdös, J. KomlⓈ and E. Szemerédi, On Turán's theorem for sparse graphs, Combinatorica 1 (4) (1981) 313–317.MATHMathSciNetGoogle Scholar
  2. [2]
    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
  3. [3]
    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
  4. [4]
    R. B. Boppana and M. M. Halldórsson, Approximating maximum independent set by excluding subgraphs, BIT 32 (1992) 180–196.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    J. A. Bondy and U. S. R. Murty, Graph Theory with Applications (Macmillan, London and Elsevier, New York, 1976).Google Scholar
  6. [6]
    Y. Caro, New Results on the Independence Number, Technical Report, Tel-Aviv University, 1979.Google Scholar
  7. [7]
    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
  8. [8]
    J. Håstad, Clique is hard to approximate within n 1−e, 37th Annual Symposium on Foundations of Computer Science, 1996, 627–636.Google Scholar
  9. [9]
    M. M. Halldórsson and J. Radhakrishnan, Improved approximations of Independent Sets in Bounded-Degree Graphs, SWAT'94, LNCS 824 (1994) 195–206.Google Scholar
  10. [10]
    M. M. Halldórsson and J. Radhakrishnan, Greed is Good: Approximating Independent Sets in Sparse and Bounded-Degree Graphs, Algorithmica 18 (1997) 145–163.MATHMathSciNetGoogle Scholar
  11. [11]
    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
  12. [12]
    G. L. Nemhauser and L. E. Trotter Jr., Vertex Packings: Structural Properties and Algorithms, Mathematical Programming 8 (1975) 232–248.MATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    J. B. Shearer, A Note on the Independence Number of Triangle-Free Graphs, Discrete Math. 46 (1983) 83–87.MATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    J. B. Shearer, A Note on the Independence Number of Triangle-Free Graphs, II, J. Combin. Ser. B 53 (1991) 300–307.MATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    J. B. Shearer, On the Independence Number of Sparse Graphs, Random Structures and Algorithms 5 (1995) 269–271.MathSciNetGoogle Scholar
  16. [16]
    P. Turán, On an extremal problem in graph theory (in Hungarian), Mat. Fiz. Lapok 48 (1941) 436–452.MATHMathSciNetGoogle Scholar
  17. [17]
    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

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Ingo Schiermeyer
    • 1
  1. 1.Lehrstuhl für Diskrete Mathematik und Grundlagen der InformatikTechnische UniversitÄt CottbusCottbusGermany

Personalised recommendations