Oriented Euler complexes and signed perfect matchings

Végh, László A.; Stengel, Bernhard von

doi:10.1007/s10107-014-0770-4

Cited by 5 publications

(5 citation statements)

References 34 publications

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“…The following proposition shows that with these labels,

P^{. 5 italicpt ℓ}

carries all the information about the unit vector game, and the polytope Q is not needed. The proposition was first stated in a dual version by Balthasar (, Lemma 4.10), and in essentially this form by Végh and von Stengel (, Proposition 1). Its proof also provides the first step of the proof of Theorem 1 below.…”

Section: Unit Vector Gamesmentioning

confidence: 99%

“…The edges in a perfect matching encode the pairs of 1s in a Gale evenness bitstring, which is completely labeled because the edges cover all nodes. Végh and von Stengel (, Theorem 12) give a near‐linear time algorithm that finds such a second perfect matching that, in addition, has opposite sign , which corresponds to a Nash equilibrium of positive index as would be found by a Lemke path (which, however, can be exponentially long). So this combinatorial problem is simpler than the problem of finding a Nash equilibrium of a bimatrix game, even though it gives rise to games that are hard to solve by the standard methods considered in Theorem 3.…”

Section: Hard‐to‐solve Bimatrix Gamesmentioning

confidence: 99%

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Unit vector games

Savani

Stengel

2016

Int J of Economic Theory

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McLennan and Tourky (2010) showed that "imitation games" provide a new view of the computation of Nash equilibria of bimatrix games with the Lemke-Howson algorithm. In an imitation game, the payoff matrix of one of the players is the identity matrix. We study the more general "unit vector games", which are already known, where the payoff matrix of one player is composed of unit vectors. Our main application is a simplification of the construction by Savani and von Stengel (2006) of bimatrix games where two basic equilibrium-finding algorithms take exponentially many steps: the Lemke-Howson algorithm, and support enumeration.

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“…The following proposition shows that with these labels,

P^{. 5 italicpt ℓ}

Section: Unit Vector Gamesmentioning

confidence: 99%

Section: Hard‐to‐solve Bimatrix Gamesmentioning

confidence: 99%

Unit vector games

Savani

Stengel

2016

Int J of Economic Theory

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“…We believe that our method is new. It avoids the use, as in [27,6,16,35], of oriented primoids or oriented duoids defined by Todd [34].…”

Section: Finding Another Colorfulmentioning

confidence: 99%

Colorful linear programming, Nash equilibrium, and pivots

Meunier

Sarrabezolles

2018

Discrete Applied Mathematics

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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.

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“…It is also possible to define a notion of orientation for primoids and duoids. It has be done in 1976 by Todd [64], and extended to oiks in 2012 by Végh and von Stengel [67]. Such approaches can be used for proving that some problems belong to the PPAD class.…”

Section: Generalized Complementarity Problemmentioning

confidence: 99%

Computing and proving with pivots

Meunier

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...

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Oriented Euler complexes and signed perfect matchings

Cited by 5 publications

References 34 publications

Unit vector games

Unit vector games

Colorful linear programming, Nash equilibrium, and pivots

Computing and proving with pivots

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