The minimum degree heuristic and the minimal triangulation process

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 2880, Issue , 2003, Pages 58-70

  Berry, Anne ^a   Heggernes, Pinar ^b   Simonet, Geneviève ^c  

^a CNRS   (France)

^b UNIVERSITY OF BERGEN   (Norway)

^c LIRMM   (France)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; GRAPH THEORY; HEURISTIC ALGORITHMS; MATRIX ALGEBRA; SURVEYING;

MINIMAL TRIANGULATIONS; MINIMUM DEGREE; SPARSE MATRICES; SPARSE MATRIX COMPUTATIONS; THEORETICAL BOUNDS;

TRIANGULATION;

EID: 0344392004     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/978-3-540-39890-5_6     Document Type: Article

References (23)

1. P. Amestoy, T. A. Davis, and I. S. Duff. An approximate minimum degree ordering algorithm. SIAM J. Matrix Anal. Appl., 17:886-905, 1996.
2. A. Berry. Désarticulation d'un graphe. PhD Dissertation, LIRMM, Montpellier, December 1998.
3. A. Berry, J. R. S. Blair, and P. Heggernes. Maximum Cardinality Search for Computing Minimal Triangulations. In L. Kucera, editor, Graph Theoretical Concepts in Computer Science - WG 2002, Springer Verlag, 2002. Lecture Notes in Computer Science.
4. A. Berry, J.-P. Bordat, P. Heggernes, G. Simonet, and Y. Villanger. A wide-range algorithm for minimal triangulation from an arbitrary ordering. Reports in Informatics 243, University of Bergen, Norway, 2003, and Research Report 02-200, LIRRM, Montpellier, France. Submitted to Journal of Algorithms, November 2002.
5. J. R. S. Blair, P. Heggernes, and J. A. Telle. A practical algorithm for making filled graphs minimal. Theoretical Computer Science, 250:124-141,2001.
6. E. Dahlhaus. Minimal elimination ordering inside a given chordal graph. In R. H. Möhring, editor, Graph Theoretical Concepts in Computer Science, pages 132-143. Springer Verlag, 1997. Lecture Notes in Computer Science 1335.
7. G.A. Dirac. On rigid circuit graphs. Anh. Math. Sem. Univ. Hamburg, 25:71-76, 1961.
8. D. R. Fulkerson and O. A. Cross. Incidence matrices and interval graphs. Pacific Journal of Math., 15:835-855, 1965.
9. J. A. George and J. W. H. Liu. The evolution of the minimum degree ordering algorithm. SIAM Review, 31:1-19, 1989.
10. P. Heggernes, S. Eisenstat, G. Kumfert, and A. Pothen. The computational complexity of the Minimum Degree algorithm. Proceedings of 14th Norwegian Computer Science Conference, NIK 2001, University of Tromsø, Norway. Also available as ICASE Report 2001-42, NASA/CR-2001-211421, NASA Langley Research Center, USA.
11. T. Kloks, D. Kratsch, and J. Spinrad. On treewidth and minimum fill-in of asteroidal triple-free graphs. Theoretical Computer Science, 175:309-335, 1997.
12. C. G. Lekkerkerker and J. Ch. Boland. Representation of a finite graph by a set of intervals on the real line. Fund. Math., 51:45-64, 1962.
13. J. W. H. Liu. Equivalent sparse matrix reorderings by elimination tree rotations. SIAM J. Sci. Stat. Comput., 9:424-444, 1988.
14. H. M. Markowitz. The elimination form of the inverse and its application to linear programming. Management Science, 3:255-269, 1957.
15. Matrix Market Web site: http://math.nist.gov/MatrixMarket/
16. T. Ohtsuki, L. K. Cheung, and T. Fujisawa. Minimal triangulation of a graph and optimal pivoting order in a sparse matrix. J. Math. Anal. Appl., 54:622-633, 1976.
17. A. Parra and P. Scheffler. How to use the minimal separators of a graph for its chordal triangulation. Proceedings of the 22nd International Colloquium on Automata, Languages and Programming (ICALP '95), Lecture Notes in Computer Science, 944:123-134, 1995.
18. S. Parter. The use of linear graphs in Gauss elimination. SIAM Review, 3:119-130, 1961.
19. B. Peyton. Minimal orderings revisited. SIAM J. Matrix Anal. Appl., 23:271-294, 2001.
20. D. J. Rose. A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations. In R. C. Read, editor, Graph Theory and Computing, pages 183-217. Academic Press, 1972.
21. D. J. Rose, R. E. Tarjan, and G. S. Lueker. Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput., 5:266-283, 1976.
22. W. F. Tinney and J. W. Walker. Direct solutions of sparse network equations by optimally ordered triangular factorization In Proceedings of the IEEE, 55:1801-1809, 1967.
23. M. Yannakakis. Computing the minimum fill-in is NP-complete. SIAM J. Alg. Disc. Meth., 2:77-79, 1981.