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14
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 20 (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.
Neighborly cubical polytopes
 Discrete & Computational Geometry
, 2000
"... Neighborly cubical polytopes exist: for any n ≥ d ≥ 2r + 2, there is a cubical whose rskeleton is combinatorially equivalent to that of the convex dpolytope Cn d ndimensional cube. This solves a problem of Babson, Billera & Chan. Kalai conjectured that the boundary ∂Cn d of a neighborly cubic ..."
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Cited by 18 (2 self)
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Neighborly cubical polytopes exist: for any n ≥ d ≥ 2r + 2, there is a cubical whose rskeleton is combinatorially equivalent to that of the convex dpolytope Cn d ndimensional cube. This solves a problem of Babson, Billera & Chan. Kalai conjectured that the boundary ∂Cn d of a neighborly cubical polytope Cn d maximizes the fvector among all cubical (d − 1)spheres with 2n vertices. While we show that this is true for polytopal spheres if n ≤ d+1, we also give a counterexample for d = 4 and n = 6. Further, the existence of neighborly cubical polytopes shows that the graph of the ndimensional cube, where n ≥ 5, is “dimensionally ambiguous ” in the sense of Grünbaum. We also show that the graph of the 5cube is “strongly 4ambiguous”. In the special case d = 4, neighborly cubical polytopes have f3 = f0 4 log2 f0 4 vertices, so the facetvertex ratio f3/f0 is not bounded; this solves a problem of Kalai, Perles and Stanley studied by Jockusch.
The random facet simplex algorithm on combinatorial cubes
 Random Structures & Algorithms
, 2001
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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 ..."
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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 &quot;geometric&quot; 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 t ..."
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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.
Polytope Skeletons And Paths
 HANDBOOK OF DISCRETE AND COMPUTATIONAL GEOMETRY (SECOND EDITION ), CHAPTER 20
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PROJECTED PRODUCTS OF POLYGONS
, 2004
"... It is an open problem to characterize the cone of fvectors of 4dimensional convex polytopes. The question whether the “fatness ” of the fvector of a 4polytope can be arbitrarily large is a key problem in this context. Here we construct a 2parameter family of 4dimensional polytopes π(P 2r n) w ..."
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Cited by 6 (1 self)
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It is an open problem to characterize the cone of fvectors of 4dimensional convex polytopes. The question whether the “fatness ” of the fvector of a 4polytope can be arbitrarily large is a key problem in this context. Here we construct a 2parameter family of 4dimensional polytopes π(P 2r n) with extreme combinatorial structure. In this family, the “fatness ” of the fvector gets arbitrarily close to 9; an analogous invariant of the flag vector, the “complexity, ” gets arbitrarily close to 16. The polytopes are obtained from suitable deformed products of even polygons by a projection to R4.
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.
On the monotone upper bound problem
, 2003
"... The Monotone Upper Bound Problem asks for the maximal number M(d, n) ofvertices on a strictlyincreasing edgepath on a simple dpolytope with n facets. More specifically, it asks whether the upper bound M(d, n) ≤ Mubt(d, n) provided by McMullen’s (1970) Upper Bound Theorem is tight, where Mubt(d, ..."
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Cited by 4 (1 self)
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The Monotone Upper Bound Problem asks for the maximal number M(d, n) ofvertices on a strictlyincreasing edgepath on a simple dpolytope with n facets. More specifically, it asks whether the upper bound M(d, n) ≤ Mubt(d, n) provided by McMullen’s (1970) Upper Bound Theorem is tight, where Mubt(d, n) is the number of vertices of a dualtocyclic dpolytope with n facets. It was recently shown that the upper bound M(d, n) ≤ Mubt(d, n) holdswith equality for small dimensions (d ≤ 4: Pfeifle, 2003) and for small corank (n ≤ d +2: Gärtner et al., 2001). Here we prove that it is not tight in general: In dimension d =6apolytopewithn =9facetscanhaveMubt(6, 9) = 30 vertices, but not more than 27 ≤ M(6, 9) ≤ 29 vertices can lie on a strictlyincreasing edgepath. The proof involves classification results about neighborly polytopes, Kalai’s (1988) concept of abstract objective functions, the HoltKlee conditions (1998), explicit enumeration, Welzl’s (2001) extended Gale diagrams, randomized generation of instances, as well as nonrealizability proofs via a version of the Farkas lemma.
Polyhedral Surfaces in Wedge Products
, 2007
"... We introduce the wedge product of two polytopes which is dual to the wreath product of Joswig and Lutz [6]. The wedge product of a pgon and a (q − 1)simplex contains many pgon faces of which we select a subcomplex corresponding to a surface. This surface is regular of type {p, 2q}, that is, all fa ..."
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Cited by 4 (0 self)
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We introduce the wedge product of two polytopes which is dual to the wreath product of Joswig and Lutz [6]. The wedge product of a pgon and a (q − 1)simplex contains many pgon faces of which we select a subcomplex corresponding to a surface. This surface is regular of type {p, 2q}, that is, all faces are pgons, all vertices have degree 2q, and the combinatorial automorphism group acts transitively on the flags of the surface. We show that for certain choices of parameters p and q there exists a realization of the wedge product such that the surface survives the projection to R 4. For a different choice of parameters such a realization does not exist.