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Randomized Simplex Algorithms on KleeMinty Cubes
 COMBINATORICA
, 1994
"... We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we use combinatorial models for the KleeMinty cubes [22] and similar linear programs with exponential decreasing paths. The analysis of two most natural randomized pivot rules on the KleeMinty cubes ..."
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Cited by 21 (6 self)
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We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we use combinatorial models for the KleeMinty cubes [22] and similar linear programs with exponential decreasing paths. The analysis of two most natural randomized pivot rules on the KleeMinty cubes leads to (nearly) quadratic lower bounds for the complexity of linear programming with random pivots. Thus we disprove two bounds (for the expected running time of the randomedge simplex algorithm on KleeMinty cubes) conjectured in the literature. At the same time, we establish quadratic upper bounds for the expected length of a path for a simplex algorithm with random pivots on the classes of linear programs under investigation. In contrast to this, we find that the average length of an increasing path in a KleeMinty cube is exponential when all paths are taken with equal probability.
The random facet simplex algorithm on combinatorial cubes
 Random Structures & Algorithms
, 2001
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Combinatorics with a geometric flavor: some examples
 in Visions in Mathematics Toward 2000 (Geometric and Functional Analysis, Special Volume
, 2000
"... In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg's theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen's upper bound t ..."
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Cited by 7 (0 self)
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In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg's theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen's upper bound theorem for the face numbers of convex polytopes and its many extensions is the second topic. Next are general properties of subsets of the vertices of the discrete ndimensional cube and some relations with questions of extremal and probabilistic combinatorics. Our fourth topic is tree enumeration and random spanning trees, and finally, some combinatorial and geometrical aspects of the simplex method for linear programming are considered.
Randomized Subexponential Algorithms for Infinite Games
, 2004
"... The complexity of solving infinite games, including parity, mean payoff, and simple stochastic games, is an important open problem in verification, automata theory, and complexity theory. In this paper we develop an abstract setting for studying and solving such games, as well as related problems, b ..."
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Cited by 6 (0 self)
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The complexity of solving infinite games, including parity, mean payoff, and simple stochastic games, is an important open problem in verification, automata theory, and complexity theory. In this paper we develop an abstract setting for studying and solving such games, as well as related problems, based on function optimization over certain discrete structures. We introduce new classes of completely localglobal (CLG) and recursively localglobal (RLG) functions, and show that strategy evaluation functions for parity and simple stochastic games belong to these classes. We also establish a relation to the previously wellstudied completely unimodal (CU) and localglobal functions. A number of nice properties of CLGfunctions are proved. In this setting, we survey several randomized optimization algorithms appropriate for CU, CLG, and RLGfunctions. We show that the subexponential algorithms for linear programming by Kalai and Matouˇsek, Sharir, and Welzl, can be adapted to optimizing the functions we study, with preserved subexponential expected running time. We examine the relations to two other abstract frameworks for subexponential
The Simplex Algorithm in Dimension Three
, 2004
"... We investigate the worstcase behavior of the simplex algorithm on linear programs with three variables, that is, on 3dimensional simple polytopes. Among the pivot rules that we consider, the “random edge” rule yields the best asymptotic behavior as well as the most complicated analysis. All other ..."
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Cited by 6 (2 self)
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We investigate the worstcase behavior of the simplex algorithm on linear programs with three variables, that is, on 3dimensional simple polytopes. Among the pivot rules that we consider, the “random edge” rule yields the best asymptotic behavior as well as the most complicated analysis. All other rules turn out to be much easier to study, but also produce worse results: Most of them show essentially worstpossible behavior; this includes both Kalai’s “randomfacet” rule, which without dimension restriction is known to be subexponential, as well as Zadeh’s deterministic historydependent rule, for which no nonpolynomial instances in general dimensions have been found so far.
Randomized Simplex Algorithms and Random Cubes (Extended Abstract)
, 1999
"... Despite its eminent practical impact, the general question for the theoretical behavior of the Simplex Algorithm is still open. However, in recent years some progress has been made by investigating randomized pivot rules. A main thread through the history of the analysis of the Simplex Algorithm has ..."
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Cited by 1 (0 self)
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Despite its eminent practical impact, the general question for the theoretical behavior of the Simplex Algorithm is still open. However, in recent years some progress has been made by investigating randomized pivot rules. A main thread through the history of the analysis of the Simplex Algorithm has been the study of linear programs on combinatorial cubes. In this paper
A Discourse on the Pivot Rules RANDOM EDGE and RANDOM FACET
, 2000
"... This technical report is a summary of most of the topics I was working on during my first year at the ETH. Hopefully, I hope to expand on some of them during my dissertation. It is, therefore, to be regarded as work in progress. The largest section is about finding new nontrivial bounds for Random E ..."
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This technical report is a summary of most of the topics I was working on during my first year at the ETH. Hopefully, I hope to expand on some of them during my dissertation. It is, therefore, to be regarded as work in progress. The largest section is about finding new nontrivial bounds for Random Edge. It opens, however, with a few facts we observed when looking at orientations on the hypercube and their impact on the behaviour of randomized pivot rules. The final section is dedicated to some ideas connected to Balinski's Theorem and other path properties on oriented polytopes. 3 Chapter 2 Randomized pivot rules on the hypercube 2.1 The Matousek class The subexponential bounds established for randomized pivot rules are valid for more general settings like the socalled LPtype problems  not just for LP as such. Most interestingly, Matousek could construct a class of problems [4] whose LP instances, as shown by Gartner [1], can be solved by Random Facet in polynomial time while some of its nonLP instances are proof that the subexponential bounds are tight. We studied how Random Edge is doing on the Matousek examples, and claim now the following: In dimension d
RANDOM EDGE can be exponential on cubes
"... We prove that RANDOM EDGE, the simplex algorithm that always chooses a random improving edge to proceed on, can take an exponential number of steps in the model of abstract objective functions (introduced by Wiliamson Hoke [27] and by Kalai [16] under different names). We define an abstract objectiv ..."
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We prove that RANDOM EDGE, the simplex algorithm that always chooses a random improving edge to proceed on, can take an exponential number of steps in the model of abstract objective functions (introduced by Wiliamson Hoke [27] and by Kalai [16] under different names). We define an abstract objective function on the ndimensional cube for which the algorithm, started at a random vertex, needs at least exp(const steps with high probability. The best previous lower bound was quadratic. So in order for RANDOM EDGE to succeed in polynomial time, geometry must help.