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Computing and proving with pivots

Published online by Cambridge University Press:  09 October 2013

Frédéric Meunier
Frédéric Meunier*
Affiliation:
UniversitéParis Est, CERMICS (ENPC), 77455 Marne-la-Vallée, France. frederic.meunier@enpc.fr

Abstract

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A simple idea used in many combinatorial algorithms is the idea ofpivoting. Originally, it comes from the method proposed by Gauss in the19th century for solving systems of linear equations. This method had been extended in1947 by Dantzig for the famous simplex algorithm used for solving linear programs. Fromsince, a pivoting algorithm is a method exploring subsets of a ground set and going fromone subset σ to a new one σ′ by deleting anelement inside σ and adding an element outside σ:σ′ = σ\ { v}  ∪  {u},with v ∈ σ and u ∉ σ.This simple principle combined with other ideas appears to be quite powerful for manyproblems. This present paper is a survey on algorithms in operations research and discretemathematics using pivots. We give also examples where this principle allows not only tocompute but also to prove some theorems in a constructive way. A formalisation isdescribed, mainly based on ideas by Michael J. Todd.

Keywords

Combinatorial topologycomplementarity problemsconstructive proofspivoting algorithms

Information

Type
Research Article
Information
RAIRO - Operations Research , Volume 47 , Issue 4 , September 2013 , pp. 331 - 360
Copyright
© EDP Sciences, ROADEF, SMAI, 2013

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