A Further Generalization of the Colourful Carathéodory Theorem

Discrete Applied Mathematics

Sarrabezolles

2018

Self Cite

The colorful Carathéodory theorem, proved by Bárány in 1982, states that given d + 1 sets of points S 1 , . . . , S d+1 in R d , with each S i containing 0 in its convex hull, there exists a set T ⊆ d+1 i=1 S i containing 0 in its convex hull and such that |T ∩ S i | ≤ 1 for all i ∈ {1, . . . , d + 1}. An intriguing question -still open -is whether such a set T , whose existence is ensured, can be found in polynomial time. In 1997, Bárány and Onn defined colorful linear programming as algorithmic questions related to the colorful Carathéodory theorem. The question we just mentioned comes under colorful linear programming.The traditional applications of colorful linear programming lie in discrete geometry. In this paper, we study its relations with other areas, such as game theory, operations research, and combinatorics. Regarding game theory, we prove that computing a Nash equilibrium in a bimatrix game is a colorful linear programming problem. We also formulate an optimization problem for colorful linear programming and show that as for usual linear programming, deciding and optimizing are computationally equivalent. We discuss then a colorful version of Dantzig's diet problem. We also propose a variant of the Bárány algorithm, which is an algorithm computing a set T whose existence is ensured by the colorful Carathéodory theorem. Our algorithm makes a clear connection with the simplex algorithm and we discuss its computational efficiency. Related complexity and combinatorial results are also provided.

Section: Main Contributions "mentioning

confidence: 93%

Section: Main Contributions "mentioning

confidence: 99%

Colorful linear programming, Nash equilibrium, and pivots

Discrete Applied Mathematics

Sarrabezolles

2018

Self Cite

“…There are examples showing that it is not sufficient that the convex hulls of triples contain the origin, but weaker relaxations are possible [274]. Arocha et al [29] also proved another "very colorful Carathéodory theorem".…”

Section: Carathéodorymentioning

confidence: 99%

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

Loera

Goaoc

et al. 2019

Bull. Amer. Math. Soc.

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“…, d+1. Meunier and Deza [45], reformulating a result of [14] (the "Octahedron Lemma"), note that the number of colorful simplices containing the origin in its convex hull is even. They describe a pivoting algorithm finding another colorful simplex containing the origin, provided there is already one.…”

Section: Scarf Lemmamentioning

confidence: 99%

Computing and proving with pivots

2013

RAIRO-Oper. Res.

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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...