Reducing the ll-in size for an elimination tree

In sparse Cholesky factorization, nding elimination orderings that produce small ll-in is very important and various heuristics have been proposed. For parallel computations, good orderings should produce elimination trees of low height. Finding optimal ll-in orderings and nding optimal height orderings are NP-hard problems. A class of Nested Dissection orderings with minimal separators, has been shown to contain optimal height orderings. We show that for such orderings, the problem of computing minimum ll-in decomposes into independent subproblems for each separator. These subproblems are shown to be NP-hard. However, for small separators, they can be solved optimally in reasonable time. We also introduce a new heuristic for these subproblems.