Advertisement

Algorithmica

, Volume 18, Issue 1, pp 145–163 | Cite as

Greed is good: Approximating independent sets in sparse and bounded-degree graphs

  • M. M. Halldórsson
  • J. Radhakrishnan
Article

Abstract

Theminimum-degree greedy algorithm, or Greedy for short, is a simple and well-studied method for finding independent sets in graphs. We show that it achieves a performance ratio of (Δ+2)/3 for approximating independent sets in graphs with degree bounded by Δ. The analysis yields a precise characterization of the size of the independent sets found by the algorithm as a function of the independence number, as well as a generalization of Turán’s bound. We also analyze the algorithm when run in combination with a known preprocessing technique, and obtain an improved\((2\bar d + 3)/5\) performance ratio on graphs with average degree\(\bar d\), improving on the previous best\((\bar d + 1)/2\) of Hochbaum. Finally, we present an efficient parallel and distributed algorithm attaining the performance guarantees of Greedy.

Key Words

Independent set problem Heuristics Approximation algorithms 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    N. Alon, U. Feige, A. Wigderson, and D. Zuckerman. Derandomized graph products.Computational Complexity, 5 (1): 60–75, 1995.zbMATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and hardness of approximation problems.Proc. 33rd Ann. IEEE Symp. on Foundations of Computer Science, pages 14–23, Oct. 1992.Google Scholar
  3. [3]
    P. Berman and T. Fujito. On the approximation properties of the independent set problem in degree 3 graphs.Proc. Fourth Workshop on Algorithms and Data Structures pages 449–460. LNCS #955, Springer-Verlag, Berlin, 1995.Google Scholar
  4. [4]
    P. Berman and M. Fürer. Approximating maximum independent set in bounded degree graphs.Proc. Fifth Ann. ACM-SIAM Symp. on Discrete Algorithms, pages 365–371, Jan. 1994.Google Scholar
  5. [5]
    R. L. Brooks. On coloring the nodes of a network.Math. Proc. Cambridge Philos. Soc., 37: 194–197, 1991.MathSciNetCrossRefGoogle Scholar
  6. [6]
    V. Chvátal.Linear Programming. Freeman, New York, 1983.zbMATHGoogle Scholar
  7. [7]
    V. Chvátal and C. McDiarmid. Small transversals in hypergraphs.Combinatorica, 12 (1): 19–26, 1992.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    P. Erdős. On the graph theorem of Turán (in Hungarian).Mat. Lapok 21: 249–251, 1970.MathSciNetGoogle Scholar
  9. [9]
    A. V. Goldberg, S. A. Plotkin, and G. E. Shannon. Parallel symmetry-breaking in sparse graphs.SIAM J. Discrete Math., 1 (4): 434–446, No. 1988.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    J. R. Griggs. Lower bounds on the independence number in terms of the degrees.J. Combin. Theory Ser. B, 34: 22–39, 1983.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    M. M. Halldórsson and J. Radhakrishnan. Improved approximations of independent sets in bounded-degree graphs.Proc. Fourth Scand. Workshop on Algorithm Theory, pages 195–206. LNCS #824, Springer-Verlag, Berlin, 1994.Google Scholar
  12. [12]
    M. M. Halldórsson and J. Radhakrishnan. Improved approximations of independent sets in bounded-degree via subgraph removal.Nordic J. Comput., 1 (4): 475–492, 1994.zbMATHMathSciNetGoogle Scholar
  13. [13]
    M. M. Halldórsson and K. Yoshihara. Greedy approximations of independent sets in low degree graphs.Proc. Sixth Internat. Symp. on Algorithms and Computation, pages 152–161. LNCS #1004, Springer-Verlag, Berlin, Dec. 1995.Google Scholar
  14. [14]
    D. S. Hochbaum. Efficient bounds for the stable set, vertex cover, and set packing problems.Discrete Appl. Math., 6: 243–254, 1983.zbMATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    J. Hopcroft and R. Karp. Ann 5/2 algorithm for maximal matchings in bipartite graphs.SIAM J. Comput., 4: 225–231, 1973.MathSciNetCrossRefGoogle Scholar
  16. [16]
    D. S. Johnson. Worst case behavior of graph coloring algorithms.Proc. 5th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, pages 513–527. Congressus Numerantium, X, Utilitas Math., Winnipeg, Manitoba, 1974.Google Scholar
  17. [17]
    R. Karp and V. Ramachandran. A survey of parallel algorithms for shared-memory machines. In J. van Leeuwen, editor,Handbook of Theoretical Computer Science, volume A, Chapter 17, pages 869–941. Elsevier, Amsterdam, 1990.Google Scholar
  18. [18]
    S. Khanna, R. Motwani, M. Sudan, and U. Vazirani. On syntactic versus computational views of approximability.Proc. 35th Ann. IEEE Symp. on Foundations of Computer Science, pages 819–830, 1994.Google Scholar
  19. [19]
    E. Kubicka, G. Kubicki, and D. Kountanis. Approximation algorithms for the chromatic sum.Proc. 1st Great Lakes Computer Science Conf. LNCS #507, Springer-Verlag, Berlin, Oct. 1989.Google Scholar
  20. [20]
    E. Lawler.Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, New York, 1976.zbMATHGoogle Scholar
  21. [21]
    N. Linial. Locality in distributive algorithms.SIAM J. Comput., 21: 193–201, 1992.zbMATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    L. Lovász. Three short proofs in graph theory.J. Combin. Theory Ser. B, 19: 269–271, 1975.zbMATHCrossRefGoogle Scholar
  23. [23]
    C. McDiarmid. Colouring random graphs.Ann. Oper. Res., 1: 183–200, 1984.zbMATHCrossRefGoogle Scholar
  24. [24]
    G. L. Nemhauser and L. Trotter. Vertex packings Structural properties and algorithms.Math. Programming, 8: 232–248, 1975.zbMATHMathSciNetCrossRefGoogle Scholar
  25. [25]
    C. H. Papadimitriou and M. Yannakakis. Optimization, approximation, and complexity classes.J. Comput. System Sci., 43: 425–440, 1991.zbMATHMathSciNetCrossRefGoogle Scholar
  26. [26]
    J. B. Shearer. A note on the independence number of triangle-free graphs.Discrete Math., 46: 83–87, 1983.zbMATHMathSciNetCrossRefGoogle Scholar
  27. [27]
    H. U. Simon. On approximate solutions for combinatorial optimization problems.SIAM J. Discrete Math., 3 (2): 294–310, May 1990.zbMATHMathSciNetCrossRefGoogle Scholar
  28. [28]
    P. Turán. On an extremal problem in graph theory (in Hungarian).Mat. Fiz. Lapok, 48: 436–452, 1941.zbMATHMathSciNetGoogle Scholar
  29. [29]
    Twentieth Century Fox.Wall Street. Motion picture, 1987.Google Scholar
  30. [30]
    V. K. Wei. A lower bound on the stability number of a simple graph. Technical Memorandum No. 81-11217-9, Bell Laboratories, 1981.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1997

Authors and Affiliations

  • M. M. Halldórsson
    • 1
  • J. Radhakrishnan
    • 2
  1. 1.Science InstituteUniversity of IcelandReykjavikIceland
  2. 2.Tata Institute of Fundamental ResearchTheoretical Computer Science GroupMumbaiIndia

Personalised recommendations