Abstract
The Minimum Degree Algorithm, one of the classical algorithms of sparse matrix computations, is a heuristic for computing a minimum triangulation of a graph. It is widely used as a component in every sparse matrix package, and it is known to produce triangulations with few fill edges in practice, although no theoretical bound or guarantee has been shown on its quality. Another interesting behavior of Minimum Degree observed in practice is that it often results in a minimal triangulation. Our goal in this paper is to examine the theoretical reasons behind this good performance. We give new invariants which partially explain the mechanisms underlying this heuristic. We show that Minimum Degree is in fact resilient to error, as even when an undesirable triangulating edge with respect to minimal triangulation is added at some step of the algorithm, at later steps the chances of adding only desirable edges remain intact. We also use our new insight to propose an improvement of this heuristic, which introduces at most as many fill edges as Minimum Degree but is guaranteed to yield a minimal triangulation.
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References
Amestoy, P., Davis, T.A., Duff, I.S.: An approximate minimum degree ordering algorithm. SIAM J. Matrix Anal. Appl. 17, 886–905 (1996)
Berry, A.: Désarticulation d’un graphe. PhD Dissertation, LIRMM, Montpellier (December 1998)
Berry, A., Blair, J.R.S., Heggernes, P.: Maximum Cardinality Search for Computing Minimal Triangulations. In: Kučera, L. (ed.) WG 2002. LNCS, vol. 2573, pp. 1–12. Springer, Heidelberg (2002)
Berry, A., Bordat, J.-P., Heggernes, P., Simonet, G., Villanger, Y.: A widerange 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)
Blair, J.R.S., Heggernes, P., Telle, J.A.: A practical algorithm for making filled graphs minimal. Theoretical Computer Science 250, 124–141 (2001)
Dahlhaus, E.: Minimal elimination ordering inside a given chordal graph. In: Möhring, R.H. (ed.) WG 1997. LNCS, vol. 1335, pp. 132–143. Springer, Heidelberg (1997)
Dirac, G.A.: On rigid circuit graphs. Anh. Math. Sem. Univ. Hamburg 25, 71–76 (1961)
Fulkerson, D.R., Gross, O.A.: Incidence matrices and interval graphs. Pacific. Journal of Math 15, 835–855 (1965)
George, J.A., Liu, J.W.H.: The evolution of the minimum degree ordering algorithm. SIAM Review 31, 1–19 (1989)
Heggernes, P., Eisenstat, S., Kumfert, G., Pothen, A.: The computational complexity of the Minimum Degree algorithm. In: Proceedings of 14th Norwegian Computer Science Conference, NIK, University of Tromsø, Norway. Also available as ICASE Report 2001-42, NASA/CR-2001-211421, NASA Langley Research Center, USA (2001)
Kloks, T., Kratsch, D., Spinrad, J.: On treewidth and minimum fill-in of asteroidal triple-free graphs. Theoretical Computer Science 175, 309–335 (1997)
Lekkerkerker, C.G., Boland, J.C.: Representation of a finite graph by a set of intervals on the real line. Fund. Math. 51, 45–64 (1962)
Liu, J.W.H.: Equivalent sparse matrix reorderings by elimination tree rotations. SIAM J. Sci. Stat. Comput. 9, 424–444 (1988)
Markowitz, H.M.: The elimination form of the inverse and its application to linear programming. Management Science 3, 255–269 (1957)
Matrix Market Web site, http://math.nist.gov/MatrixMarket/
Ohtsuki, T., Cheung, L.K., Fujisawa, T.: Minimal triangulation of a graph and optimal pivoting order in a sparse matrix. J. Math. Anal. Appl. 54, 622–633 (1976)
Parra, A., Scheffler, P.: How to use the minimal separators of a graph for its chordal triangulation. In: Fülöp, Z., Gecseg, F. (eds.) ICALP 1995. LNCS, vol. 944, pp. 123–134. Springer, Heidelberg (1995)
Parter, S.: The use of linear graphs in Gauss elimination. SIAM Review 3, 119–130 (1961)
Peyton, B.: Minimal orderings revisited. SIAM J. Matrix Anal. Appl. 23, 271–294 (2001)
Rose, D.J.: A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations. In: Read, R.C. (ed.) Graph Theory and Computing, pp. 183–217. Academic Press, London (1972)
Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5, 266–283 (1976)
Tinney, W.F., Walker, J.W.: Direct solutions of sparse network equations by optimally ordered triangular factorization. Proceedings of the IEEE 55, 1801–1809 (1967)
Yannakakis, M.: Computing the minimum fill-in is NP-complete. SIAM J. Alg. Disc. Meth. 2, 77–79 (1981)
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Berry, A., Heggernes, P., Simonet, G. (2003). The Minimum Degree Heuristic and the Minimal Triangulation Process. In: Bodlaender, H.L. (eds) Graph-Theoretic Concepts in Computer Science. WG 2003. Lecture Notes in Computer Science, vol 2880. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39890-5_6
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DOI: https://doi.org/10.1007/978-3-540-39890-5_6
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