A Spectral Algorithm for Seriation and the Consecutive Ones Problem

SIAM Journal on Computing

Volumn 28, Issue 1, 1998, Pages 297-310

  Atkins, Jonathan E ^a   Boman, Erik G ^b   Hendrickson, Bruce ^c  

^a Infinity   (United States)

^b STANFORD UNIVERSITY   (United States)

^c SANDIA NATIONAL LABORATORIES   (United States)

Author keywords

Analysis of algorithms; Consecutive ones property; Eigenvector; Fiedler vector; Seriation

Indexed keywords

CORRELATION METHODS; EIGENVALUES AND EIGENFUNCTIONS; HEURISTIC METHODS; PROBLEM SOLVING; SPECTRUM ANALYSIS;

CONSECUTIVE ONES PROBLEMS; FIEDLER VECTOR; SERIATION;

ALGORITHMS;

EID: 0032057853     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0097539795285771     Document Type: Article

References (33)

1. F. ALIZADEH, Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM J. Optim., 5 (1995), pp. 13-51.
2. N. ALON AND N. KAHALE, A spectral technique for coloring random 3-colorable graphs, in Proc. 26th Annual Symposium on Theory of Computing, ACM, New York, 1994, pp. 346-355.
3. B. ASPVALL AND J. R. GILBERT, Graph coloring using eigenvalue decomposition, SIAM J. Alg. Disc. Meth., 5 (1984), pp. 526-538.
4. S. T. BARNARD, A. POTHEN. AND H. D. SIMON, A spectral algorithm for envelope reduction of sparse matrices, in Proc. Supercomputirig '93, IEEE, Piscataway, NJ, 1993. pp. 493-502.
5. K. S. BOOTH AND G. S. LUEKER, Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms, J. Cornput. System Sci., 13 (1976), pp. 333-379.
6. F. R. K. CHUNG AND S.-T. YAU, A near optimal algorithm for edge seperators (preliminary version), in Proc. 26th Annual Symposium on Theory of Computing, ACM, New York, 1994. pp. 1-8.
7. D. M. CVETKOVIC̀, M. DOOB, I. GUTMAN, AND A. TORGASEV, Recent Results in the Theory of Graph Spectra, Annals of Discrete Mathematics 36. North-Holland, Amsterdam, 1988.
8. D. M. CVETKOVIĆ, M. DOOB, AND H. SACHS, Spectra of Graphs: Theory and Application, Academic Press, New York, 1980.
9. W. E. DONATH AND A. J. HOFFMAN. Lower bounds for the partitioning of graphs, IBM J. Res. Develop., 17 (1973), pp. 420-425.
10. M. FIEDLER, Algebraic connectivity of graphs, Czech. Math. Journal, 23 (1973), pp. 298-305.
11. M. FIEDLER. A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory, Czech. Math. Journal, 25 (1975). pp. 619-633.
12. D. R. FULKERSON AND O. A. GROSS. Incidence matrices and interval graphs. Pacific J. Math., 3 (1965), pp. 835-855.
13. A. GEORGE AND A. POTHEN, An analysis of spectral envelope-reduction via quadratic assignment problems, SLAM J. Matrix Anal. Appl., 18 (1997), pp. 706-732.
14. M. X. GOEMANS AND D. P. WlLLIAMSON, .878-approximation algorithms for MAX CUT and MAX 2SAT, in Proc. 26th Annual Symposium on Theory of Computing, ACM, New York, 1994, pp. 422-431.
15. D. S. GREENBERG AND S. C. ISTRAIL, Physical mapping by STS hybridization: Algorithmic strategies and the challenge of software evaluation, J. Comput. Biology, 2 (1995), pp. 219-274.
16. M. GRÖTSCHEL, L. LOVÁSZ, AND A. SCHRIJVER, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin, 1988.
17. M. JUVAN AND B. MOHAR, Optimal linear labelings and eigenvalues of graphs, Disc. Appl. Math., 36 (1992), pp. 153-168.
18. D. KARGER, R. MOTWANI, AND M. SUDAN, Approximate graph coloring by semidefinite programming, in Proc. 35th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1994, pp. 2-13.
19. D. G. KENDALL, Incidence matrices, interval graphs and seriation in archaeology, Pacific J. Math., 28 (1969), pp. 565-570.
20. D. G. KENDALL, Abundance matrices and seriation in archaeology, Zeitschrift für Wahrschein-lichkeitstheorie, 17 (1971), pp. 104-112.
21. D. C. KOZEN, The Design and Analysis of Algorithms, Springer-Verlag, New York, 1992.
22. M. MlGNOTTE, Identification of algebraic numbers, J. Algorithms, 3 (1982), pp. 197-204.
23. B. MOHAR, The Laplacian spectrum, of graphs, in Graph Theory, Combinatorics and Applications, Y. Alavi, G. Chartrand, O. Oellermann, and A. Schwenk, eds., Wiley, New York, 1991, pp. 871-898.
24. B. MOHAR, Laplace eigenvalues of graphs - a survey, Disc. Math., 109 (1992), pp. 171-183.
25. V. PAN, Algebraic complexity of computing polynomial zeros, Comput. Math. Appl., 14 (1987), pp. 285-304.
26. B. PARLETT AND D. SCOTT, The Lanczos algorithm with selective orthogonalization, Math. Comp., 33 (1979), pp. 217-238.
27. O. PERRON, Zur Theorie der Matrizen, Math. Ann., 1907, pp. 248-263.
28. 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.
29. W. S. ROBINSON, A method for chronologically ordering archaeological deposits, American Antiquity, 16 (1951), pp. 293-301.
30. R. S. VARGA, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.
31. M. VELDHORST, Approximation of the consecutive ones matrix augmentation problem, SIAM J. Comput., 14 (1985), pp. 709-729.
32. E. M. WILKINSON, Mathematics in the archaeological and historical sciences, in Archaeological Seriation and the Travelling Salesman Problem, University Press, Edinburgh, 1971, pp. 276-284.
33. E. M. WILKINSON, Techniques of data analysis and seriation theory, in Technische und Natur-wissenschaftliche Beiträge Zur Feldarchäologie, Rheinland-Verlag, Cologne, Germany, 1974, pp. 1-142.