A comparison of two effective methods for reordering columns within supernodes

TL;DR

This paper compares two effective methods for reordering columns within supernodes in sparse Cholesky factors, highlighting that partition refinement offers similar quality but with significantly less computational cost.

Contribution

The paper provides a fair comparison between TSP-based and partition refinement methods, demonstrating the efficiency advantages of partition refinement for supernode column reordering.

Findings

Both methods produce similar quality reorderings.

Partition refinement requires less time and storage.

Partition refinement is recommended for practical use.

Abstract

In some recent papers, researchers have found two very good methods for reordering columns within supernodes in sparse Cholesky factors; these reorderings can be very useful for certain factorization methods. The first of these reordering methods is based on modeling the underlying problem as a traveling salesman problem (TSP), and the second of these methods is based on partition refinement (PR). In this paper, we devise a fair way to compare the two methods. While the two methods are virtually the same in the quality of the reorderings that they produce, PR should be the method of choice because PR reorderings can be computed using far less time and storage than TSP reorderings.