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Spectral techniques in graph algorithms

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LATIN'98: Theoretical Informatics (LATIN 1998)

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Abstract

The existence of efficient algorithms to compute the eigenvectors and eigenvalues of graphs supplies a useful tool for the design of various graph algorithms. In this survey we describe several algorithms based on spectral techniques focusing on their performance for randomly generated input graphs.

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Research supported in part by a USA Israeli BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.

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References

  1. N. Alon, Eigenvalues and expanders, Combinatorial 6 (1986), 83–96.

    MATH  MathSciNet  Google Scholar 

  2. N. Alon and R. B. Boppana, The monotone circuit complexity of Boolean functions, Combinatorica 7 (1987), 1–22.

    MathSciNet  Google Scholar 

  3. N. Alon and N. Kahale, A spectral technique for coloring random 3-colorable graphs, Proc. of the 26th ACM STOC, ACM Press (1994), 346–355. Also; SIAM J. Comput., in press.

    Google Scholar 

  4. N. Alon and N. Kahale, Approximating the independence number via the θ-function, Math. Programming, in press.

    Google Scholar 

  5. N. Alon, M. Krivelevich and B. Sudakov, Finding a large hidden clique in a random graph, Proc. of the Ninth Annual ACM-SIAM SODA, ACM Press (1998), to appear.

    Google Scholar 

  6. N. Alon and V. D. Milman, Eigenvalues, expanders and superconcentrators, Proc. 25 th Annual Symp. on Foundations of Computer Science, Singer Island, Florida, IEEE (1984), 320–322. (Also: λ1, isoperimetric inequalities for graphs and superconcentrators, J. Combinatorial Theory, Ser. B 38 (1985), 73–88.)

    Google Scholar 

  7. N. Alon and J. H. Spencer, The Probabilistic Method, Wiley, New York, 1992.

    Google Scholar 

  8. S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy, Proof verification and intractability of approximation problems, Proc. of the 33rd IEEE FOCS, IEEE (1992), 14–23.

    Google Scholar 

  9. S. Arora and S. Safra, Probabilistic checking of proofs; a new characterization of NP, Proc. of the 33rd IEEE FOCS, IEEE (1992), 2–13.

    Google Scholar 

  10. R. Boppana and M. M. Halldórsson, Approximating maximum independent sets by excluding subgraphs, BIT, 32 (1992),180–196.

    Article  MathSciNet  Google Scholar 

  11. A. Blum, Some tools for approximate 3-coloring, Proc. 31st IEEE FOCS, IEEE (1990), 554–562.

    Google Scholar 

  12. T.N. Bui and C. Jones, Finding good approximate vertex and edge partitions is NP-hard, Infor. Proc. Letters 42 (1992), 153–159.

    Article  MathSciNet  Google Scholar 

  13. A. Blum and D. Karger, An ~O(n3/14)-colormg algorithm for 3-colorable graphs, IPL 61 (1997), 49–53.

    Article  MathSciNet  Google Scholar 

  14. B. Bollobás, Random Graphs, Academic Press, London, 1985.

    Google Scholar 

  15. A. Blum and J. H. Spencer, Coloring random and semi-random k-colorable graphs, Journal of Algorithms 19 (1995), 204–234.

    Article  MathSciNet  Google Scholar 

  16. R. Boppana, Eigenvalues and graph bisection: An average case analysis, Proc. 28th IEEE FOCS, IEEE (1987), 280–285.

    Google Scholar 

  17. F. Chung and S. T. Yau, Eigenvalues, flows and separators of graphs, to appear.

    Google Scholar 

  18. M. E. Dyer and A. M. Frieze, The solution of some random NP-Hard problems in polynomial expected time, Journal of Algorithms 10 (1989), 451–489.

    Article  MathSciNet  Google Scholar 

  19. W. E. Donath and A. J. Hoffman, Lower bounds for the partitioning of graphs, J. Res. Develop. 17 (1973), 420–425.

    MathSciNet  Google Scholar 

  20. P. Erdós and A. Rényi, On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci. 5 (1960), 17–61.

    Google Scholar 

  21. M. Fiedler, Algebraic connectivity of graphs, Czechoslovak Math. J. 23(98) (1973), 298–305.

    MATH  MathSciNet  Google Scholar 

  22. U. Feige, S. Goldwasser, L. Lovász, S. Safra and M. Szegedy, Approximating Clique is almost NP-complete, Proc. of the 32nd IEEE FOCS, IEEE (1991), 2–12.

    Google Scholar 

  23. Z. Füredi and J. Komlós, The eigenvalues of random symmetric matrices, Combinatorica 1 (1981), 233–241.

    MathSciNet  Google Scholar 

  24. U. Feige and J. Kilian, Zero knowledge and the chromatic number, Proc. 11th Annual IEEE Conf. on Computational Complexity, 1996.

    Google Scholar 

  25. J. Friedman, J. Kahn and E. Szemerédi, On the second eigenvalue in random regular graphs, Proc. 21st ACM STOC, ACM Press (1989), 587–598.

    Google Scholar 

  26. A. Frieze and C. McDiarmid, Algorithmic theory of random graphs, Random Structures and Algorithms 10 (1997), 5–42.

    Article  MathSciNet  Google Scholar 

  27. M. R. Garey and D. S. Johnson, Computers and intractability: a guide to the theory of NP-completeness, Freeman and Company, 1979.

    Google Scholar 

  28. J. Håstad, Clique is hard to approximate within n 1−∃, Proc. 37th IEEE FOCS, IEEE (1996), 627–636.

    Google Scholar 

  29. A. J. Hoffman, On eigenvalues and colorings of graphs, in: B. Harris Ed., Graph Theory and its Applications, Academic, New York and London, 1970, 79–91.

    Google Scholar 

  30. M. Jerrum, Large cliques elude the metropolis process, Random Structures and Algorithms 3 (1992), 347–359.

    MATH  MathSciNet  Google Scholar 

  31. R. M. Karp, Reducibility among combinatorial problems, In: Complexity of computer computations, R. E. Miller and J. W. Thatcher (eds.), Plenum Press, New York, 1972, pp. 85–103.

    Google Scholar 

  32. R. M. Karp, Probabilistic analysis of some combinatorial search problems, In: Algorithms and Complexity: New Directions and Recent Results, J. F. Traub, ed., Academic Press, New York, 1976, pp. 1–19.

    Google Scholar 

  33. L. Kučera, Expected behavior of graph colouring algorithms, In Lecture Notes in Computer Science No. 56, Springer-Verlag, 1977, pp. 447–451.

    Google Scholar 

  34. L. Kučera, Expected complexity of graph partitioning problems, Discrete Applied Math. 57 (1995), 193–212.

    Article  Google Scholar 

  35. D. Karger, R. Motwani, and M. Sudan. Approximate graph coloring by semi-definite programming, In 35th Symposium on Foundations of Computer Science, pages 2–13. IEEE Computer Society Press, 1994.

    Google Scholar 

  36. L. Lovász, On the Shannon capacity of a graph, IEEE Transactions on Information Theory IT-25, (1979), 1–7.

    Article  Google Scholar 

  37. L. Lovász, Combinatorial Problems and Exercises, North Holland, Amsterdam, 1979, Chapter 11.

    Google Scholar 

  38. F. T. Leighton and S. Rao, An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms, Proc 29th annual FOCS (1988), 422–431.

    Google Scholar 

  39. R. J. Lipton and R. E. Tarjan, A separator theorem for planar graphs, SIAM J. Appl. Math. 36(1979), 177–189.

    Article  MathSciNet  Google Scholar 

  40. C. Lund and M. Yannakakis, On the hardness of approximating minimization problems. Proc. 25th ACM STOC (1993), 286–293.

    Google Scholar 

  41. D. W. Matula, On the complete subgraph of a random graph, Combinatory Mathematics and its Applications, Chapel Hill, North Carolina (1970), 356–369.

    Google Scholar 

  42. A. Pothem, H. D. Simon and K.-P. Liou, Partitioning sparse matrices with eigenvectors of graphs, SIAM J. Matrix Anal. Appl. 11 (1990), 430–452.

    Article  MathSciNet  Google Scholar 

  43. A. D. Petford and D. J. A. Welsh, A randomised 3-colouring algorithm, Discrete Mathematics, 74 (1989), 253–261.

    Article  MathSciNet  Google Scholar 

  44. A. Ralston, A First Course in Numericad Analysis, McGraw-Hill, 1985, Section 10.4.

    Google Scholar 

  45. A. A. Razborov, Lower bounds for the monotone complexity of some Boolean functions, Dokl. Ak. Nauk. SSSR 281 (1985), 798–801 (in Russian). English translation in: Sov. Math. Dokl. 31 (1985), 354–357.

    MATH  MathSciNet  Google Scholar 

  46. M. Saks, Private communication.

    Google Scholar 

  47. H. D. Simon, Partitioning of unstructured problems for parallel processing, Computing Systmes in Engineering 2(2/3) (1991), 135–148.

    Article  Google Scholar 

  48. A. Sinclair and M. R. Jerrum, Approximate counting, uniform generation and rapidly mixing Markov chains, Information and Computation 82 (1989), 93–133.

    Article  MathSciNet  Google Scholar 

  49. D. A. Spielman and S.-H. Teng, Spectral partitioning works: planar graphs and finite element meshes, to appear.

    Google Scholar 

  50. J. S. Turner, Almost all k-colorable graphs are easy to color, Journal of Algorithms 9 (1988), 63–82.

    Article  MATH  MathSciNet  Google Scholar 

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Cláudio L. Lucchesi Arnaldo V. Moura

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© 1998 Springer-Verlag Berlin Heidelberg

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Alon, N. (1998). Spectral techniques in graph algorithms. In: Lucchesi, C.L., Moura, A.V. (eds) LATIN'98: Theoretical Informatics. LATIN 1998. Lecture Notes in Computer Science, vol 1380. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054322

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  • DOI: https://doi.org/10.1007/BFb0054322

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  • Online ISBN: 978-3-540-69715-2

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