# Decomposing Matrices into Blocks

@article{Borndrfer1998DecomposingMI, title={Decomposing Matrices into Blocks}, author={Ralf Bornd{\"o}rfer and Carlos Eduardo Ferreira and Alexander Martin}, journal={SIAM J. Optim.}, year={1998}, volume={9}, pages={236-269} }

In this paper we investigate whether matrices arising from linear or integer programming problems can be decomposed into so-called bordered block diagonal form. More precisely, given some matrix A, we try to assign as many rows as possible to some number $\beta$ of blocks of size $\kappa$ such that no two rows assigned to different blocks intersect in a common column. Bordered block diagonal form is desirable because it can guide and speed up the solution process for linear and integer…

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## References

SHOWING 1-10 OF 32 REFERENCES

### Sparse Matrix Ordering Methods for Interior Point Linear Programming

- Computer ScienceINFORMS J. Comput.
- 1998

A new method, based on the nested dissection heuristic, provides significantly better orderings than the most commonly used ordering method, minimum degree, on a variety of large-scale linear programming problems.

### The node capacitated graph partitioning problem: A computational study

- Computer ScienceMath. Program.
- 1998

A variety of separation heuristics for cycle, cycle with ears, knapsack tree and path-block cycle inequalities among others are presented and a formulation including variables for the edges with nonzero costs and node partition variables is presented.

### Solving Multiple Knapsack Problems by Cutting Planes

- Computer ScienceSIAM J. Optim.
- 1996

The inequalities that are described here serve as the theoretical basis for a cutting plane algorithm that is applied to practical problem instances arising in the design of main frame computers, in the layout of electronic circuits, and in sugar cane alcohol production.

### Theory of linear and integer programming

- MathematicsWiley-Interscience series in discrete mathematics and optimization
- 1999

Introduction and Preliminaries. Problems, Algorithms, and Complexity. LINEAR ALGEBRA. Linear Algebra and Complexity. LATTICES AND LINEAR DIOPHANTINE EQUATIONS. Theory of Lattices and Linear…

### PARTITIONING SPARSE MATRICES WITH EIGENVECTORS OF GRAPHS*

- Computer Science
- 1973

It is shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph, which can be used to compute good separators in grid graphs.

### Parallel Algorithms for Matrix Computations

- Computer Science
- 1987

This book consists of two comprehensive survey papers on important parallel algorithms for solving problems arising in the major areas of numerical linear algebra direct solution of linear systems, least squares computations, eigenvalue and singular value Computations, and rapid elliptic solvers.

### Geometric Algorithms and Combinatorial Optimization

- Mathematics
- 1981

0. Mathematical Preliminaries.- 0.1 Linear Algebra and Linear Programming.- Basic Notation.- Hulls, Independence, Dimension.- Eigenvalues, Positive Definite Matrices.- Vector Norms, Balls.- Matrix…

### Facets and lifting procedures for the set covering polytope

- MathematicsMath. Program.
- 1989

The strong connections between several combinatorial problems and the covering problem are pointed out and, exploiting those connections, some examples are presented of new facets for the Knapsack and Acyclic Subdigraph polytopes.

### Packing Steiner trees: a cutting plane algorithm and computational results

- Computer ScienceMath. Program.
- 1996

This paper uses a cutting plane algorithm based on the polyhedral theory for the Steiner tree packing polyhedron to solve some switchbox routing problems of VLSI-design and reports on the computational experience.

### On the facial structure of set packing polyhedra

- MathematicsMath. Program.
- 1973

This paper shows that the cliques of the intersection graph provide a first set of facets for the polyhedron in question, and it is shown that the cycles without chords of odd length of the intersections graph give rise to a further set of facet.