Lemke Paths on Simple Polytopes

Morris, Jr. Walter D.

doi:10.1287/moor.19.4.780

Cited by 16 publications

(32 citation statements)

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“…This is always so in our special case (3) where q = 1, C = −M, and P is bounded. For a simple polytope P in (3), a "Lemke path" (see also Morris, 1994) is obtained for a given missing label w in [m] as follows. Start at a CL vertex, for example 0, and "pivot" along the unique edge that leaves the facet with label w. This reaches a vertex v on a new facet F with label k. If k = w, then v is CL and the path terminates.…”

Section: Labeled Polytopes and Signed Matchingsmentioning

confidence: 99%

“…The main result of this section (Theorem 12) states that in an Euler digraph, a second perfect matching of opposite sign can be found in polynomial (in fact, near-linear) time. This holds in contrast to the complementary pivoting algorithm, which can take exponential time; Casetti, Merschen, and von Stengel (2010) have shown how to apply results of Morris (1994) for this purpose. However, the pivoting algorithm takes linear time in a bipartite Euler graph, and a variant can be used to find an oppositely signed matching in a bipartite graph that has no source or sink (Proposition 13).…”

Section: Signed Perfect Matchingsmentioning

confidence: 99%

“…The construction is adapted from the exponentially long Lemke paths of Morris (1994) for labeled dual cyclic polytopes. The completely labeled vertices of such polytopes correspond to perfect matchings in Euler graphs, as noted by Casetti, Merschen, and von Stengel (2010), in the following way.…”

Section: Proposition 13 Consider a Bipartite Graph G = (V E) With Anmentioning

confidence: 99%

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Oriented Euler complexes and signed perfect matchings

Végh

Stengel

2014

Math. Program.

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This paper presents "oriented pivoting systems" as an abstract framework for complementary pivoting. It gives a unified simple proof that the endpoints of complementary pivoting paths have opposite sign. A special case are the Nash equilibria of a bimatrix game at the ends of Lemke-Howson paths, which have opposite index. For Euler complexes or "oiks", an orientation is defined which extends the known concept of oriented abstract simplicial manifolds. Ordered "room partitions" for a family of oriented oiks come in pairs of opposite sign. For an oriented oik of even dimension, this sign property holds also for unordered room partitions. In the case of a two-dimensional oik, these are perfect matchings of an Euler graph, with the sign as defined for Pfaffian orientations of graphs. A near-linear time algorithm is given for the following problem: given a graph with an Eulerian orientation with a perfect matching, find another perfect matching of opposite sign. In contrast, the complementary pivoting algorithm for this problem may be exponential.

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Section: Labeled Polytopes and Signed Matchingsmentioning

confidence: 99%

Section: Signed Perfect Matchingsmentioning

confidence: 99%

Section: Proposition 13 Consider a Bipartite Graph G = (V E) With Anmentioning

confidence: 99%

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Oriented Euler complexes and signed perfect matchings

Végh

Stengel

2014

Math. Program.

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“…Morris [47] have found a infinite family of polytopes for which the number of pivots performed by the Lemke algorithm is exponential. More recently, a family of bimatrix games instances leading also to an exponential number of pivots has been found by Savani and von Stengel [55].…”

Section: 2mentioning

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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“…Savani and von Stengel [8] (see also [6]) have shown that the Lemke-Howson algorithm is exponential relative to the size of the matrices defining the game, not necessarily exponential relative to the number of simplices defined by the matrices. Proof.…”

Section: Nash Equilibria For 2-person Gamesmentioning

confidence: 99%

On Finding Another Room-Partitioning of the Vertices

Edmonds

Sanità

2010

Electronic Notes in Discrete Mathematics

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Let T be a triangulated surface given by the list of vertex-triples of its triangles, called rooms. A room-partitioning of T is a subset R of the rooms such that each vertex of T is in exactly one room in R.We prove that if T has a room-partitioning R, then there is another roompartitioning of T which is different from R. The proof is a simple algorithm which walks from room to room, which however we show to be exponential by constructing a sequence of (planar) instances, where the algorithm walks from room to room an exponential number of times relative to the number of rooms in the instance.We unify the above theorem with Nash's theorem stating that a 2-person game has an equilibrium, by proving a combinatorially simple common generalization.

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Lemke Paths on Simple Polytopes

Cited by 16 publications

References 12 publications

Oriented Euler complexes and signed perfect matchings

Oriented Euler complexes and signed perfect matchings

Computing and proving with pivots

On Finding Another Room-Partitioning of the Vertices

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