Combinatorial Problems in Solving Linear Systems

Abstract

Abstract. Numerical linear algebra and combinatorial optimization are vast subjects; as is their interaction. In virtually all cases there should be a notion of sparsity for a combinatorial problem to arise. Sparse matrices therefore form the basis of the interaction of these two seemingly disparate subjects. As the core of many of today's numerical linear algebra computations consists of the solution of sparse linear system by direct or iterative methods, we survey some combinatorial problems, ideas, and algorithms relating to these computations. On the direct methods side, we discuss issues such as matrix ordering; bipartite matching and matrix scaling for better pivoting; task assignment and scheduling for parallel multifrontal solvers. On the iterative method side, we discuss preconditioning techniques including incomplete factorization preconditioners, support graph preconditioners, and algebraic multigrid. In a separate part, we discuss the block triangular form of sparse matrices.

Approximation algorithms in combinatorial scientific computing

“…Their code has been incorporated into the distributed-memory SuperLU solver for sparse, unsymmetric, linear systems of equations (Li and Demmel 2003). Additional details on the applications of matchings in sparse matrix algorithms is provided in Duff and Uçar (2012).…”

Constructing Elimination Trees for Sparse Unsymmetric Matrices

“…It is used to estimate and optimize the storage and computational requirements during symbolic and numerical factorizations [10]. The elimination tree for symmetric (positive definite) systems dates from 80s [11] and even before (see [4,Section 3.3] and [10]), whereas the elimination tree for unsymmetric matrices is a recent development. Eisenstat and Liu [6] define the elimination tree for unsymmetric matrices and discuss its properties.…”

Computing symmetric nonnegative rank factorizations

“…We next note that property ∆ max is equivalent to the existence of an independent set of size r in the graph of the matrix [17]. Moreover, this is a maximum independent set, since the existence of an independent set of size larger than r, say r + 1, would imply the existence of a principal diagonal submatrix of size r + 1 so the rank of the matrix would be at least r + 1, thus violating property ∆ max .…”