# Making an arbitrary filled graph minimal by removing fill edges

Conference paper

First Online:

## Abstract

We consider the problem of removing fill edges from a filled graph *G*^{′} to get a minimal chordal supergraph *M* of the original graph *G*; thus *G*\(\subseteq\)*M*\(\subseteq\)*G*^{′}. We show that a greedy strategy can be applied if fill edges are processed for removal in the reverse order of their introduction. For a filled graph with *f* fill edges and *e* original edges, we give a simple *O*(*f*(*e* + *f*)) algorithm which solves the problem and computes a corresponding minimal elimination ordering. We believe that in practice the runtime of our algorithm is usually better than the worst-case bound of *O*(*f*(*e*+*f*)).

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.A.Agrawal, P.Klein, and R.Ravi. Cutting down on fill using nested dissection: provably good elimination orderings. In J. A. George, J. R. Gilbert, and J. W. H. Liu, editors,
*Sparse matrix computations: Graph theory issues and algorithms*, pages 31–55. Springer Verlag, 1993. IMA Volumes in Mathematics and its Applications, Vol. 56.Google Scholar - 2.C. Beeri, R. Fagin, D. Maier, and M. Yannakakis. On the desirability of acyclic database systems.
*J. Assoc. Comput. Mach.*, 30:479–513, 1983.Google Scholar - 3.F. R. K. Chung and D. Mumford. Chordal completions of planar graphs.
*J. Comb. Theory*, 31:96–106, 1994.Google Scholar - 4.E. Dahlhaus and M. Karpinski. An efficient parallel algorithm for the minimal elimination ordering of an arbitrary graph.
*Proceedings FOCS*, pages 454–459, 1989.Google Scholar - 5.R. E. England, J. R. S. Blair, and M. G. Thomason. Independent computations in a probablistic knowledge-based system. Technical Report CS-90-128, Department of Computer Science, The University of Tennessee, Knoxville, Tennessee, 1991.Google Scholar
- 6.A. George and J.W-H. Liu.
*Computer Solution of Large Sparse Positive Definite Systems*. Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1981.Google Scholar - 7.S. L. Lauritzen and D. J. Spiegelhalter. Local computations with probabilities on graphical structures and their applications to expert systems.
*J. Royal Statist. Soc., ser B*, 50:157–224, 1988.Google Scholar - 8.J. G. Lewis, B. W. Peyton, and A. Pothen. A fast algorithm for reordering sparse matrices for parallel factorization.
*SIAM J. Sci. Stat. Comput.*, 10:1156–1173, 1989.Google Scholar - 9.J. W-H. Liu and A. Mirzaian. A linear reordering algorithm for parallel pivoting of chordal graphs.
*SIAM J. Disc. Math.*, 2:100–107, 1989.Google Scholar - 10.T. Ohtsuki, L.K. Cheung, and T. Fujisawa. Minimal triangulation of a graph and optimal pivoting ordering in a sparse matrix.
*J. Math. Anal. Appl.*, 54:622–633, 1976.Google Scholar - 11.B. W. Peyton.
*Some applications of clique trees to the solution of sparse linear systems*. PhD thesis, Dept. of Mathematical Sciences, Clemson University, 1986.Google Scholar - 12.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.Google Scholar - 13.D.J. Rose, R.E. Tarjan, and G.S. Lueker. Algorithmic aspects of vertex elimination on graphs.
*SIAM J. Comput.*, 5:266–283, 1976.CrossRefGoogle Scholar - 14.R. E. Tarjan and M. Yannakakis. Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs.
*SIAM J. Comput.*, 13:566–579, 1984.CrossRefGoogle Scholar - 15.M. Yannakakis. Computing the minimum fill-in is NP-complete.
*SIAM J. Alg. Disc. Meth.*, 2:77–79, 1981.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1996