Simple and Efficient Modifications of Elimination Orderings

One of the most important and well studied problems related to sparse Cholesky factorization is to compute elimination orderings that give as few nonzero entries as possible in the resulting factors. We study the problem of modifying a given elimination ordering through local reorderings. We present new theoretical results on equivalent orderings, including a new characterization of such orderings. Based on these results, we define the notion of k-optimality for an elimination ordering, and we describe how to use this in a practical context to modify a given elimination ordering to obtain less fill. We experiment with different values of k, and report on percentage of fill that is actually reduced from an already good initial ordering, like Minimum Degree.