Ranks and representations for spectral graph bisection

SIAM Journal on Scientific Computing

Volumn 31, Issue 5, 2009, Pages 3529-3546

  Martin, Jacob G ^a   Canfield, E Rodney ^b  

^a GEORGETOWN UNIVERSITY MEDICAL CENTER   (United States)

^b University of Georgia ^*  (United States)

Author keywords

Adjacency representations; Graph bisection; Graph partitioning; Minimum graph bisection; Spectral analysis; Spectral bisection; Spectral methods

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; GRAPH THEORY; GRAPHIC METHODS; LAPLACE TRANSFORMS; MATRIX ALGEBRA;

ADJACENCY MATRIX; ADJACENCY REPRESENTATION; GRAPH BISECTION; GRAPH PARTITIONING; LAPLACIANS; MINIMUM GRAPH BISECTION; MINIMUM GRAPHS; SPECTRAL BISECTION; SPECTRAL METHODS;

SPECTRUM ANALYSIS;

EID: 79960115446     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/070710640     Document Type: Article

References (41)

1. N. Alon, Spectral techniques in graph algorithms (invited paper), in Proceedings of LATIN'98: Theoretical Informatics (Campinas, 1998), Lecture Notes in Comput. Sci. 1380, Springer, Berlin, 1998, pp. 206-215. (Pubitemid 128056278)
2. C. J. Alpert and S.-Z. Yao, Spectral partitioning: The more eigenvectors, the better, in IEEE Design Automation Conference, IEEE, Washington, DC, 1995, pp. 195-200.
3. B. Aspvall and J. R. Gilbert, Graph coloring using eigenvalue decomposition, SIAM J. Algebraic Discrete Methods, 5 (1984), pp. 526-538.
4. E. R. Barnes, An algorithm for partitioning the nodes of a graph, SIAM J. Algebraic Discrete Methods, 3 (1982), pp. 541-550.
5. R. B. Boppana, Eigenvalues and graph bisection: An average-case analysis (extended abstract), in Proceedings of FOCS, IEEE Computer Society, Washington, DC, 1987, pp. 280-285.
6. T. N. Bui, S. Chaudhuri, F. T. Leighton, and M. Sipser, Graph bisection algorithms with good average case behavior, Combinatorica, 7 (1987), pp. 171-191.
7. T. N. Bui and C. Jones, Finding good approximate vertex and edge partitions is NP-hard, Inform. Process. Lett., 42 (1992), pp. 153-159.
8. T. N. Bui and C. Jones, A heuristic for reducing fill-in in sparse matrix factorization, in Proceedings of PPSC, SIAM, Philadelphia, 1993, pp. 445-452.
9. T. N. Bui and B. R. Moon, Genetic algorithm and graph partitioning, IEEE Trans. Comput., 45 (1996), pp. 841-855. (Pubitemid 126784444)
10. P. Chardaire, M. Barake, and G. P. McKeown, A probe-based heuristic for graph partitioning, IEEE Trans. Comput., 56 (2007), pp. 1707-1720. (Pubitemid 350141889)
11. D. M. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs, Pure and Applied Mathematics 87, Academic Press, New York, 1980.
12. J. Day, W. So, and R. C. Thompson, The spectrum of a Hermitian matrix sum, Linear Algebra Appl., 280 (1998), pp. 289-332.
13. W. E. Donath and A. J. Hoffman, Lower bounds for the partitioning of graphs, IBM J. Res. Develop., 17 (1973), pp. 420-425.
14. P. Erdós and A. Rényi, On the evolution of random graphs, Bull. Inst. Internat. Statist., 38 (1961), pp. 343-347.
15. C. M. Fiduccia and R. M. Mattheyses, A linear-time heuristic for improving network partitions, in Proceedings of the 19th Conference on Design Automation (Piscataway, NJ, 1982), IEEE Press, Washington, DC, 1982, pp. 175-181.
16. M. Fiedler, Algebraic connectivity of graphs, Czechoslovak Math. J., 23 (1973), pp. 298-305.
17. A. Frieze and C. McDiarmid, Algorithmic theory of random graphs, Random Structures Algorithms, 10 (1997), pp. 5-42.
18. M. R. Garey, D. S. Johnson, and L. Stockmeyer, Some simplified NP-complete problems, in Proceedings of the Sixth Annual ACM Symposium on Theory of Computing, ACM, New York, 1974, pp. 47-63.
19. M. R. Garey, D. S. Johnson, and L. Stockmeyer, Some simplified NP-complete graph problems, Theoret. Comput. Sci., 1 (1976), pp. 237-267.
20. S. Guattery and G. L. Miller, On the quality of spectral separators, SIAM J. Matrix Anal. Appl., 19 (1998), pp. 701-719.
21. S. Guattery and G. L. Miller, On the performance of spectral graph partitioning methods, in Proceedings of SODA, SIAM, Philadelphia, 1995, pp. 233-242.
22. B. Hendrickson and R. W. Leland, A multi-level algorithm for partitioning graphs, in Proceedings of the 1995 ACM/IEEE Conference on Supercomputing, ACM, New York, IEEE, Washington, DC, 1995, p. 28.
23. D. S. Johnson, C. R. Aragon, L. A. McGeoch, and C. Schevon, Optimization by simulated annealing: An experimental evaluation. I. Graph partitioning, Oper. Res., 37 (1989), pp. 865-892. (Pubitemid 20625067)
24. C. A. Jones, Vertex and Edge Partitions of Graphs, Ph.D. thesis, Pennsylvania State University, University Park, PA, 1992.
25. R. M. Karp, Reducibility among combinatorial problems, in Complexity of Computer Computations (Yorktown Heights, NY, 1972), Plenum, New York, 1972, pp. 85-103.
26. B. Kernighan and S. Lin, An efficient heuristic procedure for partitioning graphs, Bell Systems J., 49 (1972), pp. 291-307.
27. J. F. Monahan, Numerical Methods of Statistics, Cambridge University Press, New York, 2001.
28. B. Monien, The bandwidth minimization problem for caterpillars with hair length 3 is NP-complete, SIAM J. Algebraic Discrete Methods, 7 (1986), pp. 505-512.
29. S. Oliveira, D. Stewart, and T. Soma, Semidefinite programming for graph partitioning with preferences in data distribution, in High Performance Computing for Computational Science, Lecture Notes in Comput. Sci. 2565, Springer, Berlin, 2003, pp. 307-321.
30. C. H. Papadimitriou, A. A. Schäffer, and M. Yannakakis, On the complexity of local search, in Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, ACM, New York, 1990, pp. 438-445.
31. C. H. Papadimitriou and M. Sideri, The bisection width of grid graphs, Math. Systems Theory, 29 (1996), pp. 97-110. (Pubitemid 126646868)
32. A. Pothen, H. D. Simon, and K.-P. Liou, Partitioning sparse matrices with eigenvectors of graphs, SIAM J. Matrix Anal. Appl., 11 (1990), pp. 430-452.
33. S. Rajendran, Computing the lowest eigenvalue with Rayleigh quotient iteration, J. Sound Vibration, 254 (2002), pp. 599-612. (Pubitemid 35275034)
34. J. E. Savage and M. G. Wloka, Parallelism in graph-partitioning, J. Parallel Distrib. Comput., 13 (1991), pp. 257-272.
35. J. J. Seidel, A survey of two-graphs, in Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973), Tomo I, Accad. Naz. Lincei, Rome, 1976. Atti dei Convegni Lincei 17, pp. 481-511.
36. A. J. Soper, C. Walshaw, and M. Cross, A combined evolutionary search and multilevel optimisation approach to graph-partitioning, J. Global Optimization, 29 (2004), pp. 225-241.
37. D. A. Spielman and S. H. Teng, Spectral partitioning works: Planar graphs and finite element meshes, Linear Algebra Appl., 421 (2007), pp. 284-305. (Pubitemid 46096985)
38. L. N. Trefethen and D. Bau, III, Numerical Linear Algebra, SIAM, Philadelphia, 1997.
39. E. R. van Dam and W. H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl., 373 (2003), pp. 241-272.
40. J. H. van Lint and J. J. Seidel, Equilateral point sets in elliptic geometry, Nederl. Akad. Wetensch. Proc. Ser. A 69=Indag. Math., 28 (1966), pp. 335-348.
41. U. von Luxburg, A tutorial on spectral clustering, Stat. Comput., 17 (2007), pp. 395-416.