Bit 34 (1994), 000{000. 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 eeective, measure of the time needed to perform parallel elimination. Finding orderings that produce low elimination trees is therefore important. As the problem of nding minimum height elimination tree orderings is NP-hard, it is interesting to concentrate on limited classes of graphs and nd minimum height elimination trees for these eeciently. In this paper, we use clique trees to nd an eecient algorithm for interval graphs which make an important subclass of chordal graphs. We rst illustrate this method through an algorithm that nds minimum height elimination trees for chordal graphs. This algorithm, although of exponential time complexity, is conceptionally simple and leads to a polynomial-time algorithm for nding minimum height elimination trees for interval graphs.