### Abstract

We consider a graph-theoretic elimination process which is related to performin~ Gaussian elimination on sparse symmetric and unsymmetrlc systems of linear equations. We discuss good al~orlthms for finding elimination orderings, showing that a generalization of breadth-flrst search, called lexicographlc search, can be used to find perfect orderlngs in O(n+e) time and minimal orderlngs in O(ne) time, if the problem graph is undirected and has n vertices and e edges. We also give efficient (though slower) algorithms for generating such orderings on directed graphs. We claim that the minimum ordering problem for directed graphs is NP-complete, and conjecture that it is also NP-complete for undirected graphs. We include a brief discussion of the relation of elimination to transitive closure and discuss some unresolved, more general, issues.

Original language | English (US) |
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Pages (from-to) | 245-254 |

Number of pages | 10 |

Journal | Proceedings of the Annual ACM Symposium on Theory of Computing |

State | Published - May 5 1975 |

Externally published | Yes |

Event | 7th Annual ACM Symposium on Theory of Computing, STOC 1975 - Albuquerque, United States Duration: May 5 1975 → May 7 1975 |

### All Science Journal Classification (ASJC) codes

- Software

### Keywords

- Breadth-flrst search
- Directed graph
- Gaussian elimination
- Graph
- Lexicographlc ordering
- Perfect elimination
- Sparse linear equations
- Triangulated graph

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## Cite this

*Proceedings of the Annual ACM Symposium on Theory of Computing*, 245-254.