Finding minimum height elimination trees for interval graphs in polynomial time

The elimination tree plays an important role in many aspects of sparse matrix factorization. The height of the elimination tree presents a rough, but usually effective, measure of the time needed to perform parallel elimination. Finding orderings that produce low elimination is therefore important. As the problem of finding minimum height elimination tree orderings is NP-hard, it is interesting to concentrate on limited classes of graphs and find minimum height elimination trees for these efficiently. In this paper, we use clique trees to find an efficient algorithm for interval graphs which make an important subclass of chordal graphs. We first illustrate this method through an algorithm that finds minimum height elimination for chordal graphs. This algorithm, although of exponential time complexity, is conceptionally simple and leads to a polynomial-time algorithm for finding minimum height elimination trees for interval graphs.

[1]  G. Dirac.  On rigid circuit graphs , 1961 .

[2]  P. Gilmore,et al.  A Characterization of Comparability Graphs and of Interval Graphs , 1964, Canadian Journal of Mathematics.

[3]  D. R. Fulkerson,et al.  Incidence matrices and interval graphs , 1965 .

[4]  E. Cuthill,et al.  Reducing the bandwidth of sparse symmetric matrices , 1969, ACM '69.

[5]  N. Sloane,et al.  Proof Techniques in Graph Theory , 1970 .

[6]  Ronald C. Read,et al.  Graph theory and computing , 1972 .

[7]  D. Rose.  A GRAPH-THEORETIC STUDY OF THE NUMERICAL SOLUTION OF SPARSE POSITIVE DEFINITE SYSTEMS OF LINEAR EQUATIONS , 1972 .

[8]  F. Gavril.  The intersection graphs of subtrees in tree are exactly the chordal graphs , 1974 .

[9]  R. Tarjan.  Graph theory and Gaussian elimination. , 1975 .

[10]  Kellogg S. Booth,et al.  Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-Tree Algorithms , 1976, J. Comput. Syst. Sci..

[11]  M. Golumbic.  CHAPTER 3 – Perfect Graphs , 1980 .

[12]  Philip A. Bernstein,et al.  Power of Natural Semijoins , 1981, SIAM J. Comput..

[13]  M. Yannakakis.  Computing the Minimum Fill-in is NP^Complete , 1981 .

[14]  Robert E. Tarjan,et al.  Simple Linear-Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs , 1984, SIAM J. Comput..

[15]  M. Heath.  Sparse matrix computations , 1984, The 23rd IEEE Conference on Decision and Control.

[16]  R. Möhring.  Algorithmic graph theory and perfect graphs , 1986 .

[17]  Richard C. T. Lee,et al.  Counting Clique Trees and Computing Perfect Elimination Schemes in Parallel , 1989, Inf. Process. Lett..

[18]  Joseph W. H. Liu.  The role of elimination trees in sparse factorization , 1990 .

[19]  Peter L. Hammer,et al.  Difference graphs , 1990, Discret. Appl. Math..

[20]  Pinar Heggernes.  Minimizing Fill-in Size and Elimination Tree Height in Parallel Cholesky Factorization , 1992 .

[21]  B. Peyton,et al.  An Introduction to Chordal Graphs and Clique Trees , 1993 .

[22]  A. George,et al.  Graph theory and sparse matrix computation , 1993 .

[23]  Barry W. Peyton,et al.  On Finding Minimum-Diameter Clique Trees , 1994, Nord. J. Comput..

[24]  Jitender S. Deogun,et al.  On Vertex Ranking for Permutations and Other Graphs , 1994, STACS.