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Approximating maximum independent sets by excluding subgraphs

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Abstract

An approximation algorithm for the maximum independent set problem is given, improving the best performance guarantee known toO(n/(logn)2). We also obtain the same performance guarantee for graph coloring. The results can be combined into a surprisingly strongsimultaneous performance guarantee for the clique and coloring problems.

The framework ofsubgraph-excluding algorithms is presented. We survey the known approximation algorithms for the independent set (clique), coloring, and vertex cover problems and show how almost all fit into that framework. We show that among subgraph-excluding algorithms, the ones presented achieve the optimal asymptotic performance guarantees.

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References

  1. M. Ajtai, J. Komlós, and E. Szemerédi.A note on Ramsey numbers. J. Combin. Theory Ser A, 29: 354–360, 1980.

    Google Scholar 

  2. S. Arora and S. Safra.Approximating clique is NP-complete. Manuscript.

  3. R. Bar-Yehuda and S. Even.A 2 — loglogn/2 logn performance ratio for the weighted vertex cover problem. Technical Report #260, Technion, Haifa, Jan. 1983.

  4. B. Berger and J. Rompel.A better performance guarantee for approximate graph coloring. Algorithmica, 5 (4): 459–466, 1990.

    Google Scholar 

  5. P. Berman and G. Schnitger.On the complexity of approximating the independent set problem. Information and Computation, 96 (1): 77–94, Jan. 1992.

    Google Scholar 

  6. A. Blum.An O (n 4)approximation algorithm for 3-coloring. In Proc. 22nd Ann. ACM Symp. on Theory of Computing, pages 535–542, 1989. Submitted to J. ACM.

  7. A. Blum.Some tools for approximate 3-coloring. In Proc. 31st Ann. IEEE Symp. on Found. of Comp. Sci., pages 554–562, Oct. 1990.

  8. B. Bollobás.Random Graphs. Academic Press, 1985.

  9. R. B. Boppana and M. M. Halldórsson.Approximating maximum independent sets by excluding subgraphs. In Proc. of 2nd Scand. Workshop on Algorithm Theory. Lecture Notes in Computer Science #447, pages 13–25. Springer-Verlag, July 1990.

  10. V. Chvátal.Determining the stability number of a graph. SIAM J. Comput., 6 (4), Dec. 1977.

  11. P. Erdös.Some remarks on chromatic graphs. Colloq. Math., 16: 253–256, 1967.

    Google Scholar 

  12. P. Erdös and G. Szekeres.A combinatorial problem in geometry. Compositio Math., 2: 463–470, 1935.

    Google Scholar 

  13. U. Feige, S. Goldwasser, L. Lovász, S. Safra and M. Szegedy.Approximating clique is almost NP-complete. In Proc. 32nd Ann. IEEE Symp. on Found. of Comp. Sci., pp. 2–12, Oct. 1991.

  14. T. Gallai.Kritische graphen I. Publ. Math. Inst. Hungar. Acad. Sci., 8: 165–192, 1963. (See Bollobás, B.Extremal Graph Theory., Academic Press, 1978, page 285).

    Google Scholar 

  15. M. R. Garey and D. S. Johnson.Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, 1979.

  16. M. M. Halldórsson.A still better performance guarantee for approximate graph coloring. Technical Report 90-44, DIMACS, June 1990.

  17. D. S. Johnson.Worst case behaviour of graph coloring algorithms. In Proc. 5th Southeastern Conf. on Combinatorics, Graph Theory, and Computing. Congressus Numerantium X, pages 513–527, 1974.

  18. N. Linial and U. Vazirani.Graph products and chromatic numbers. In Proc. 30th Ann. IEEE Symp. on Found. of Comp. Sci., pages 124–128, 1989.

  19. B. Monien and E. Speckenmeyer.Ramsey numbers and an approximation algorithm for the vertex cover problem. Acta Inf., 22: 115–123, 1985.

    Google Scholar 

  20. J. B. Shearer.A note on the independence number of triangle-free graphs. Discrete Math., 46: 83–87, 1983.

    Google Scholar 

  21. A. Wigderson.Improving the performance guarantee for approximate graph coloring. J. ACM, 30 (4): 729–735, 1983.

    Google Scholar 

  22. S. Arora, C. Lund, R. Motwani, M. Sudan and M. Szegedy.On the intractability of approximation problems. Manuscript.

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A preliminary version of this paper appeared in [9].

Supported in part by National Science Foundation Grant CCR-8902522 and PYI Award CCR-9057488.

Research done at Rutgers University. Supported in part by Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) fellowship.

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Boppana, R., Halldórsson, M.M. Approximating maximum independent sets by excluding subgraphs. BIT 32, 180–196 (1992). https://doi.org/10.1007/BF01994876

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