Computing and proving with pivots

A simple idea used in many combinatorial algorithms is the idea of pivoting. Originally, it comes from the method proposed by Gauss in the 19th century for solving systems of linear equations. This method had been extended in 1947 by Dantzig for the famous simplex algorithm used for solving linear programs. From since, a pivoting algorithm is a method exploring subsets of a ground set and going from one subset σ to a new one σ′ by deleting an element 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 many problems. This present paper is a survey on algorithms in operations research and discrete mathematics using pivots. We give also examples where this principle allows not only to compute but also to prove some theorems in a constructive way. A formalisation is described, mainly based on ideas by Michael J. Todd.