On the minimum FLOPs problem in the sparse Cholesky factorization

TL;DR

This paper demonstrates that minimizing fill-in and FLOPs in sparse Cholesky factorization are fundamentally different problems and proves that minimizing FLOPs is NP-hard, challenging common assumptions.

Contribution

The paper provides a rigorous theoretical distinction between fill-in minimization and FLOP minimization, and establishes the NP-hardness of minimizing FLOPs in Cholesky factorization.

Findings

Fill-in minimization and FLOP minimization are different problems.

Minimizing FLOPs is NP-hard.

Explicit construction showing the strict difference between the two problems.

Abstract

Prior to computing the Cholesky factorization of a sparse, symmetric positive definite matrix, a reordering of the rows and columns is computed so as to reduce both the number of fill elements in Cholesky factor and the number of arithmetic operations (FLOPs) in the numerical factorization. These two metrics are clearly somehow related and yet it is suspected that these two problems are different. However, no rigorous theoretical treatment of the relation of these two problems seems to have been given yet. In this paper we show by means of an explicit, scalable construction that the two problems are different in a very strict sense. In our construction no ordering, that is optimal for the fill, is optimal with respect to the number of FLOPs, and vice versa. Further, it is commonly believed that minimizing the number of FLOPs is no easier than minimizing the fill (in the complexity sense), but so far no proof appears to be known. We give a reduction chain that shows the NP hardness of minimizing the number of arithmetic operations in the Cholesky factorization.