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17
Linear Programming, the Simplex Algorithm and Simple Polytopes
 Math. Programming
, 1997
"... In the first part of the paper we survey some farreaching applications of the basic facts of linear programming to the combinatorial theory of simple polytopes. In the second part we discuss some recent developments concerning the simplex algorithm. We describe subexponential randomized pivot ru ..."
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Cited by 22 (1 self)
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In the first part of the paper we survey some farreaching applications of the basic facts of linear programming to the combinatorial theory of simple polytopes. In the second part we discuss some recent developments concerning the simplex algorithm. We describe subexponential randomized pivot rules and upper bounds on the diameter of graphs of polytopes. 1 Introduction: A convex polyhedron is the intersection P of a finite number of closed halfspaces in R d . P is a ddimensional polyhedron (briefly, a dpolyhedron) if the points in P affinely span R d . A convex ddimensional polytope (briefly, a dpolytope) is a bounded convex dpolyhedron. Alternatively, a convex dpolytope is the convex hull of a finite set of points which affinely span R d . A (nontrivial) face F of a dpolyhedron P is the intersection of P with a supporting hyperplane. F itself is a polyhedron of some lower dimension. If the dimension of F is k we call F a kface of P . The empty set and P itself are...
The random facet simplex algorithm on combinatorial cubes
 Random Structures & Algorithms
, 2001
"... ..."
Combinatorial linear programming: Geometry can help
 Proc. 2nd Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM), Lecture Notes in Computer Science 1518
, 1998
"... We consider a class A of generalized linear programs on the dcube (due to Matousek) and prove that Kalai's subexponential simplex algorithm RandomFacet is polynomial on all actual linear programs in the class. In contrast, the subexponential analysis is known to be best possible for general inst ..."
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Cited by 9 (2 self)
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We consider a class A of generalized linear programs on the dcube (due to Matousek) and prove that Kalai's subexponential simplex algorithm RandomFacet is polynomial on all actual linear programs in the class. In contrast, the subexponential analysis is known to be best possible for general instances in A. Thus, we identify a "geometric" property of linear programming that goes beyond all abstract notions previously employed in generalized linear programming frameworks, and that can be exploited by the simplex method in a nontrivial setting.
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 theorem for ..."
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Cited by 8 (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.
The WorstCase Running Time of the Random Simplex Algorithm is Exponential in the Height
, 1995
"... The random simplex algorithm for linear programming proceeds as follows: at each step, it moves from a vertex v of the polytope to a randomly chosen neighbor of v, the random choice being made from those neighbors of v that improve the objective function. We exhibit a polytope defined by n constrai ..."
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Cited by 7 (0 self)
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The random simplex algorithm for linear programming proceeds as follows: at each step, it moves from a vertex v of the polytope to a randomly chosen neighbor of v, the random choice being made from those neighbors of v that improve the objective function. We exhibit a polytope defined by n constraints in three dimensions with height O(log n), for which the expected running time of the random simplex algorithm is \Omega\Gamma n= log n). Keywords: Linear programming, simplex algorithm, randomized algorithm We consider linear programming problems defined by n constraints involving d variables. The constraints define a polytope in d dimensions; we seek a point of the polytope that maximizes an objective function that is linear in the variables. It is known that the optimum, if bounded, occurs at a vertex of this polytope. The classic simplex algorithm begins at a vertex of the polytope and, at each step proceeds to a neighboring vertex that improves the objective function. The simplex al...
The KleeMinty random edge chain moves with linear speed
, 2004
"... Abstract: An infinite sequence of 0’s and 1’s evolves by flipping each 1 to a 0 exponentially at rate one. When a 1 flips, all bits to its right also flip. Starting from any configuration with finitely many 1’s to the left of the origin, we show that the leftmost 1 moves right with linear speed. Upp ..."
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Cited by 7 (1 self)
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Abstract: An infinite sequence of 0’s and 1’s evolves by flipping each 1 to a 0 exponentially at rate one. When a 1 flips, all bits to its right also flip. Starting from any configuration with finitely many 1’s to the left of the origin, we show that the leftmost 1 moves right with linear speed. Upper and lower bounds are given on the speed.
One Line and n Points
 Proc. 33rd Ann. ACM Symp. on the Theory of Computing (STOC
, 2003
"... We analyze a randomized pivoting process involving one line and n points in the plane. The process models the behavior of the RandomEdge simplex algorithm on simple polytopes with n facets in dimension n  2. We obtain a tight O(log² n) bound for the expected number of pivot steps. This is the ..."
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Cited by 7 (2 self)
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We analyze a randomized pivoting process involving one line and n points in the plane. The process models the behavior of the RandomEdge simplex algorithm on simple polytopes with n facets in dimension n  2. We obtain a tight O(log² n) bound for the expected number of pivot steps. This is the first nontrivial bound for RandomEdge which goes beyond bounds for specific polytopes. The process itself can be interpreted as a simple algorithm for certain 2variable linear programming problems, and we prove a tight Θ(n) bound for its expected runtime.