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Bounds on the Chvátal Rank of Polytopes in the 0/1Cube
"... Gomory's and Chvatal's cuttingplane procedure proves recursively the validity of linear inequalities for the integer hull of a given polyhedron. The number of rounds needed to obtain all valid inequalities is known as the Chvatal rank of the polyhedron. It is wellknown that the Chvatal rank can be ..."
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Cited by 28 (1 self)
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Gomory's and Chvatal's cuttingplane procedure proves recursively the validity of linear inequalities for the integer hull of a given polyhedron. The number of rounds needed to obtain all valid inequalities is known as the Chvatal rank of the polyhedron. It is wellknown that the Chvatal rank can be arbitrarily large, even if the polyhedron is bounded, if it is of dimension 2, and if its integer hull is a 0/1polytope. We prove that the Chvatal rank of polyhedra featured in common relaxations of many combinatorial optimization problems is rather small; in fact, the rank of any polytope contained in the ndimensional 0/1cube is at most 3n² lg n. This improves upon a recent result of Bockmayr et al. [6] who obtained an upper bound of O(n³ lg n). Moreover, we refine this result by showing that the rank of any polytope in the 0/1cube that is defined by inequalities with small coe#cients is O(n). The latter observation explains why for most cutting planes derived in polyhedral st...
The Many Facets of Linear Programming
, 2000
"... . We examine the history of linear programming from computational, geometric, and complexity points of view, looking at simplex, ellipsoid, interiorpoint, and other methods. Key words. linear programming  history  simplex method  ellipsoid method  interiorpoint methods 1. Introduction A ..."
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. We examine the history of linear programming from computational, geometric, and complexity points of view, looking at simplex, ellipsoid, interiorpoint, and other methods. Key words. linear programming  history  simplex method  ellipsoid method  interiorpoint methods 1. Introduction At the last Mathematical Programming Symposium in Lausanne, we celebrated the 50th anniversary of the simplex method. Here, we are at or close to several other anniversaries relating to linear programming: the sixtieth of Kantorovich's 1939 paper on "Mathematical Methods in the Organization and Planning of Production" (and the fortieth of its appearance in the Western literature) [55]; the fiftieth of the historic 0th Mathematical Programming Symposium that took place in Chicago in 1949 on Activity Analysis of Production and Allocation [64]; the fortyfifth of Frisch's suggestion of the logarithmic barrier function for linear programming [37]; the twentyfifth of the awarding of the 1975 Nobe...
Lectures on 0/1polytopes
 Polytopes — combinatorics and computation (Oberwolfach, 1997), volume 29 of DMV Seminar
, 2000
"... These lectures on the combinatorics and geometry of 0/1polytopes are meant as an introduction and invitation. Rather than heading for an extensive survey on 0/1polytopes I present some interesting aspects of these objects; all of them are related to some quite recent work and progress. 0/1polytope ..."
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Cited by 16 (1 self)
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These lectures on the combinatorics and geometry of 0/1polytopes are meant as an introduction and invitation. Rather than heading for an extensive survey on 0/1polytopes I present some interesting aspects of these objects; all of them are related to some quite recent work and progress. 0/1polytopes have a very simple definition and explicit descriptions; we can enumerate and analyze small examples explicitly in the computer (e. g. using polymake). However, any intuition that is derived from the analysis of examples in “low dimensions” will miss the true complexity of 0/1polytopes. Thus, in the following we will study several aspects of the complexity of higherdimensional 0/1polytopes: the doublyexponential number of combinatorial types, the number of facets which can be huge, and the coefficients of defining inequalities which sometimes turn out to be extremely large. Some of the effects and results will be backed by proofs in the course of these lectures; we will also be able to verify some of them on explicit examples, which are accessible as a polymake database.
Random Walks, Totally Unimodular Matrices, and a Randomised Dual Simplex Algorithm
, 2001
"... We discuss the application of random walks to generating a random basis of a totally unimodular matrix and to solving a linear program with such a constraint matrix. We also derive polynomial upper bounds on the combinatorial diameter of an associated polyhedron. ..."
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Cited by 15 (3 self)
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We discuss the application of random walks to generating a random basis of a totally unimodular matrix and to solving a linear program with such a constraint matrix. We also derive polynomial upper bounds on the combinatorial diameter of an associated polyhedron.
Polytope Skeletons And Paths
 Handbook of Discrete and Computational Geometry (Second Edition ), chapter 20
"... INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent i ..."
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INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent if they form the endpoints of an edge of P . In this chapter, we will describe results and problems concerning graphs and skeletons of polytopes. In Section 17.1 we briefly describe the situation for 3polytopes. In Section 17.2 we consider general properties of polytopal graphs subgraphs and induced subgraphs, connectivity and separation, expansion, and other properties. In Section 17.3 we discuss problems related to diameters of polytopal graphs in connection with the simplex algorithm and t
On the Expansion of Graphs of 0/1Polytopes
 The Sharpest Cut: The Impact of Manfred Padberg and His Work
, 2001
"... The edge expansion of a graph is the minimum quotient of the number of edges in a cut and the size of the smaller one among the two node sets separated by the cut. Bounding the edge expansion from below is important for bounding the \mixing time" of a random walk on the graph from above. It has been ..."
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Cited by 6 (2 self)
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The edge expansion of a graph is the minimum quotient of the number of edges in a cut and the size of the smaller one among the two node sets separated by the cut. Bounding the edge expansion from below is important for bounding the \mixing time" of a random walk on the graph from above. It has been conjectured by Mihail and Vazirani (see [9]) that the graph of every 0/1polytope has edge expansion at least one. A proof of this (or even a weaker) conjecture would imply solutions of several longstanding open problems in the theory of randomized approximate counting. We present dierent techniques for bounding the edge expansion of a 0/1polytope from below. By means of these tools we show that several classes of 0/1polytopes indeed have graphs with edge expansion at least one. These classes include all 0/1polytopes of dimension at most ve, all simple 0/1polytopes, all hypersimplices, all stable set polytopes, and all (perfect) matching polytopes.
The complexity of generic primal algorithms for solving general integer programs
 MATH. OPER. RES
, 2001
"... Primal methods constitute a common approach to solving (combinatorial) optimization problems. Starting from a given feasible solution, they successively produce new feasible solutions with increasingly better objective function value until an optimal solution is reached. From an abstract point of vi ..."
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Cited by 5 (2 self)
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Primal methods constitute a common approach to solving (combinatorial) optimization problems. Starting from a given feasible solution, they successively produce new feasible solutions with increasingly better objective function value until an optimal solution is reached. From an abstract point of view, an augmentation problem is solved in each iteration. That is, given a feasible point, these methods find an augmenting vector, if one exists. Usually, augmenting vectors with certain properties are sought to guarantee the polynomial running time of the overall algorithm. In this paper, we show that one can solve every integer programming problem in polynomial time provided one can efficiently solve the directed augmentation problem. The directed augmentation problem arises from the ordinary augmentation problem by splitting each direction into its positive and its negative part and by considering linear objectives on these parts. Our main result states that in order to get a polynomialtime algorithm for optimization it is sufficient to efficiently find, for any linear objective function in the positive and negative part, an arbitrary augmenting vector. This result also provides a general framework for the design of polynomialtime algorithms for specific combinatorial optimization problems. We demonstrate its applicability by considering the mincost
On the Relative Complexity of 15 Problems Related to 0/1Integer Programming
"... An integral part of combinatorial optimization and computational complexity consists of establishing relationships between different problems or different versions of the same problem. In this chapter, we bring together known and new, previously published and unpublished results, which establish th ..."
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Cited by 4 (1 self)
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An integral part of combinatorial optimization and computational complexity consists of establishing relationships between different problems or different versions of the same problem. In this chapter, we bring together known and new, previously published and unpublished results, which establish that 15 problems related to optimizing a linear function over a 0/1polytope are polynomialtime equivalent. This list of problems includes optimization and augmentation, testing optimality and primal separation, sensitivity analysis and inverse optimization, and several others.
Simple 0/1polytopes
 EUROPEAN J. COMBINATORICS
, 1999
"... For general polytopes, it has turned out that with respect to many questions it su ces to consider only the simple polytopes, i.e., ddimensional polytopes where every vertex is contained in only d facets. In this paper, we show that the situation is very di erent within the class of 0/1polytopes, ..."
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Cited by 4 (1 self)
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For general polytopes, it has turned out that with respect to many questions it su ces to consider only the simple polytopes, i.e., ddimensional polytopes where every vertex is contained in only d facets. In this paper, we show that the situation is very di erent within the class of 0/1polytopes, since every simple 0/1polytope is the (cartesian) product of some 0/1simplices (which proves a conjecture of Ziegler), and thus, the restriction to simple 0/1polytopes leaves only a very small class of objects with a rather trivial structure.