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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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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.
The random facet simplex algorithm on combinatorial cubes
 Random Structures & Algorithms
, 2001
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Unique sink orientations of grids
 Proc. 11th Conference on Integer Programming and Combinatorial Optimization (IPCO
, 2005
"... We introduce unique sink orientations of grids as digraph models for many wellstudied problems, including linear programming over products of simplices, generalized linear complementarity problems over Pmatrices (PGLCP), and simple stochastic games. We investigate the combinatorial structure of su ..."
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We introduce unique sink orientations of grids as digraph models for many wellstudied problems, including linear programming over products of simplices, generalized linear complementarity problems over Pmatrices (PGLCP), and simple stochastic games. We investigate the combinatorial structure of such orientations and develop randomized algorithms for finding the sink. We show that the orientations arising from PGLCP satisfy the HoltKlee condition known to hold for polytope digraphs, and we give the first expected lineartime algorithms for solving PGLCP with a fixed number of blocks.
On LPorientations of cubes and crosspolytopes
, 2002
"... Abstract. In a paper presented at a 1996 conference, Holt and Klee introduced a set of necessary conditions for an orientation of the graph of a dpolytope to be induced by a realization into R n and linear functional on that space. In general, it is an open question to decide whether for a polytope ..."
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Abstract. In a paper presented at a 1996 conference, Holt and Klee introduced a set of necessary conditions for an orientation of the graph of a dpolytope to be induced by a realization into R n and linear functional on that space. In general, it is an open question to decide whether for a polytope P every orientation of its graph satisfying these conditions can in fact be realized in this fashion; two natural families of polytopes to consider are cubes and crosspolytopes. For cubes, we show that, as n grows, the percentage of ncube HoltKlee orientations which can be realized goes asymptotically to 0. For crosspolytopes, we give a stronger set of conditions which are both necessary and sufficient for an orientation to be in this class; as a corollary, we prove that all shellings of cubes are line shellings. 1.
Monotone paths on polytopes
 MATH. Z. 235,315–334 (2000)
, 2000
"... We investigate the vertexconnectivity of the graph of fmonotone paths on a dpolytope P with respect to a generic functional f. The third author has conjectured that this graph is always (d − 1)connected. We resolve this conjecture positively for simple polytopes and show that the graph is 2co ..."
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Cited by 9 (5 self)
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We investigate the vertexconnectivity of the graph of fmonotone paths on a dpolytope P with respect to a generic functional f. The third author has conjectured that this graph is always (d − 1)connected. We resolve this conjecture positively for simple polytopes and show that the graph is 2connected for any dpolytope with d ≥ 3. However,we disprove the conjecture in general by exhibiting counterexamples for each d ≥ 4 in which the graph has a vertex of degree two. We also reexamine the Baues problem for cellular strings on polytopes, solved by Billera,Kapranov and Sturmfels. Our analysis shows that their positive result is a direct consequence of shellability of polytopes and is therefore less related to convexity than is at first apparent.
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.
Polytope Skeletons And Paths
 HANDBOOK OF DISCRETE AND COMPUTATIONAL GEOMETRY (SECOND EDITION ), CHAPTER 20
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The HoltKlee condition for oriented matroids
 European J. Combinatorics
, 2009
"... Holt and Klee have recently shown that every (generic) LP orientation of the graph of a dpolytope satisfies a directed version of the dconnectivity property, i.e. there are d internally disjoint directed paths from a unique source to a unique sink. We introduce two new classes HK and HK * of orien ..."
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Holt and Klee have recently shown that every (generic) LP orientation of the graph of a dpolytope satisfies a directed version of the dconnectivity property, i.e. there are d internally disjoint directed paths from a unique source to a unique sink. We introduce two new classes HK and HK * of oriented matroids (OMs) by enforcing this property and its dual interpretation in terms of line shellings, respectively. Both classes contain all representable OMs by the HoltKlee theorem. While we give a construction of an infinite family of nonHK * OMs, it is not clear whether there exists any nonHK OM. This leads to a fundamental question as to whether the HoltKlee theorem can be proven combinatorially by using the OM axioms only. Finally, we give the complete classification of OM(4, 8), the OMs of rank 4 on 8element ground set with respect to the HK, HK*, Euclidean and Shannon properties. Our classification shows that there exists no nonHK OM in this class. 1
Two new bounds for the randomedge simplex algorithm
, 2008
"... We prove that the RandomEdge simplex algorithm requires an expected number of at most 13n / √ d pivot steps on any simple dpolytope with n vertices. This is the first nontrivial upper bound for general polytopes. We also describe a refined analysis that potentially yields much better bounds for s ..."
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We prove that the RandomEdge simplex algorithm requires an expected number of at most 13n / √ d pivot steps on any simple dpolytope with n vertices. This is the first nontrivial upper bound for general polytopes. We also describe a refined analysis that potentially yields much better bounds for specific classes of polytopes. As one application, we show that for combinatorial dcubes, the trivial upper bound of 2 d on the performance of RandomEdge can asymptotically be improved by any desired polynomial factor in d.
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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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.