Experimental Evaluation of the Greedy and Random Algorithms for Finding Independent Sets in Random Graphs

  • M. Goldberg
  • D. Hollinger
  • M. Magdon-Ismail
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3503)


This work is motivated by the long-standing open problem of designing a polynomial-time algorithm that with high probability constructs an asymptotically maximum independent set in a random graph. We present the results of an experimental investigation of the comparative performance of several efficient heuristics for constructing maximal independent sets. Among the algorithms that we evaluate are the well known randomized heuristic, the greedy heuristic, and a modification of the latter which breaks ties in a novel way. All algorithms deliver on-line upper bounds on the size of the maximum independent set for the specific input-graph. In our experiments, we consider random graphs parameterized by the number of vertices n and the average vertex degree d. Our results provide strong experimental evidence in support of the following conjectures:

  1. 1

    for d = c · n (c is a constant), the greedy and random algorithms are asymptotically equivalent;

  2. 2

    for fixed d, the greedy algorithms are asymptotically superior to the random algorithm;

  3. 3

    for graphs with d ≤ 3, the approximation ratio of the modified greedy algorithm is asymptotically < 1.005.

We also consider random 3-regular graphs, for which non-trivial lower and upper bounds on the size of a maximum independent set are known. Our experiments suggest that the lower bound is asymptotically tight.


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  1. 1.
    Aronson, J., Frieze, A.M., Pitel, B.G.: Maximum matchings in sparse random graphs: Karp-Sipser re-visited. Random Structures and Algorithms, 111–178 (1998)Google Scholar
  2. 2.
    Bollobás, B.: Random Graphs, 2nd edn. Cambridge University Press, New York (2001)zbMATHGoogle Scholar
  3. 3.
    Bollobás, B., Erdös, P.: Cliques in random graphs. Mathematical Proceedings of the Cambridge Philosophical Society 80, 419–427 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Boppana, R., Halldórsson, M.: Approximating maximum independent sets by excluding subgraphs. BIT 20, 180–196 (1992)CrossRefGoogle Scholar
  5. 5.
    Erdös, P., Rényi, A.: On the evolution of random graphs. Magyar Tud. Akad. Mat. Kut. Int. Közl 4, 17–61 (1960)Google Scholar
  6. 6.
    Frieze, A.M., Read, B.: Probabilistic analysis of algorithms. In: Probabilistic Methods for Algorithmic Discrete Mathematics, pp. 36–92 (1998)Google Scholar
  7. 7.
    Frieze, A.M., Suen, S.: On the independence number of random cubic graphs. Random Structures and Algorithms 5, 649–664 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Halldórsson, M., Radhakrishnan, J.: Greed is good: Approximating independent sets in sparce and bounded-degree graphs. Algorithmica 18, 145–163 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Karp, R.M.: Reducibility among combinatorial problems. Complexity of Computer Computatiolns, 85–103 (1972)Google Scholar
  10. 10.
    Karp, R.M., Sipser, M.: Maximum matchings in sparse graphs. In: Proceedings of the 22nd Annual IEEE Symposium on Foundations of Computing, pp. 364–375 (1982)Google Scholar
  11. 11.
    Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. In: Proceedings of 25th Annual ACM Symp. on Theory of Computing, pp. 286–293 (1993)Google Scholar
  12. 12.
    Matula, D.: The largest clique size in a random graph. Southern Methodist University, Tech. Report, CS 7608 (1976)Google Scholar
  13. 13.
    West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice Hall, New York (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • M. Goldberg
    • 1
  • D. Hollinger
    • 1
  • M. Magdon-Ismail
    • 1
  1. 1.Computer Science DepartmentRensselaer Polytechnic Institute 

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