Computing and proving with pivots

Abstract

Abstract. 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.1. Introduction 1.1. Motivation. Pivoting is one of the oldest ideas used for an algorithm. The Gauss elimination method aims at solving linear systems of the form Ax = b. This method uses pivots. The matrix A is progressively transformed into an upper triangular matrix with 1's on the diagonal. At any step of the algorithm, the pivot is the nonzero entry on the diagonal of the current matrix used to annul the entries below it. A similar idea is used in the simplex algorithm solving linear programs. In the simplex algorithm, during the pivot operation, a column is added to a set called a basis and another column leaves the basis, leading to a fixed cardinality for the basis. In this framework, pivoting becomes a combinatorial operation. During the last decades, similar algorithms, maintaining a set of fixed size with elements entering and leaving, have been succesfully applied for other problems in mathematical programming. Their success relies on the simplicity of this idea, and also on some special structures of the problems that have been progressively revealed. Pivoting algorithms also appears naturally in some constructive proofs. One of the most famous example is Sperner's lemma providing a simple constructive proof of Brouwer's theorem and which, by a slight adaptation by Scarf, becomes a fully algorithmic proof of that same theorem with the help of pivot operations.The purpose of our paper is to provide a survey of these various applications and to describe a simple framework inspired mainly by the work by Todd [63] and in which it is quite easy to describe the ideas of pivoting. This framework may help researchers to build their own pivoting algorithms for problems they meet. Some related open questions are also stated. 1.2.2. Simplicial complexes. A useful notion in the context of algorithms using pivot operations is that of (abstract) simplicial complex. An abstract simplicial complex K is a collection of subsets of a given ground set V called its vertex set such that, for any σ ∈ K and τ ⊆ σ, we have also τ ∈ K. It implies in particular that ∅ ∈ K. An element of K is cal...

Colorful linear programming, Nash equilibrium, and pivots

“…This point of view goes back to Lemke and Howson [20]. A complete proof within this framework can be found in Remark 6.1 of [24]. We derive the difficulty of Finding Another Colorful Simplex from the complexity of Bimatrix.…”

“…The triangulation K here can easily be encoded by a Turing machine computing the neighbors of any simplex in the triangulation in polynomial time. There is a proof of Sperner's lemma (Theorem 2) via an oriented path-following argument [29,24], which considers directed paths joining fullylabeled simplices. Given a fully-labeled simplex, finding another fully-labeled simplex is thus in PPAD, and so is Finding Another Colorful Simplex.…”

Small Degenerate Simplices Can Be Bad for Simplex Methods

“…In addition, Zadeh pointed out that his constructions, and many others requiring an exponential number of iterations, occur in so-called deformed products of polytopes. For more details about pivot based algorithms, instances requiring an exponential number of iterations for simplex methods, and related results, we refer to the surveys of Meunier [7], Terlaky and Zhang [8], and Ziegler [10], and to the recent results of Avis and Friedmann [2], and references therein.…”