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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.
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.
The random facet simplex algorithm on combinatorial cubes
 Random Structures & Algorithms
, 2001
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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.
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.
Polytope Skeletons And Paths
 HANDBOOK OF DISCRETE AND COMPUTATIONAL GEOMETRY (SECOND EDITION ), CHAPTER 20
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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 6 (4 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.
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.
On Combinatorial Properties of Linear Program Digraphs
, 2008
"... The possible pivot operations of the simplex method to solve a linear program can be represented as a directed graph defined on the skeleton of the feasible region P. We consider the case that P is bounded, i.e., a convex polytope. The directed graph is called an LP digraph. LP digraphs are known to ..."
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The possible pivot operations of the simplex method to solve a linear program can be represented as a directed graph defined on the skeleton of the feasible region P. We consider the case that P is bounded, i.e., a convex polytope. The directed graph is called an LP digraph. LP digraphs are known to satisfy the following three properties: acyclicity, unique sink orientation(USO), and the HoltKlee property. The three properties are not generally sufficient for a directed graph on the skeleton of P to be an LP digraph. In this paper, we first survey some previous results on LP digraphs, showing relationships among the three properties. Then we introduce a new necessary property for a directed graph on the skeleton of P to be an LP digraph, called the shelling property. We analyze the relationships between the shelling property and the three existing properties, showing that it is stronger than a combination of acyclicity and USO for nonsimple polytopes in dimension at least four. In all other cases it is equivalent to the intersection of these two properties. 1
Long Monotone Paths on Simple 4Polytopes
"... of vertices in a monotone path along edges of a ddimensional polytope with n facets can be as large as conceivably possible: Is M(d, n) = Mubt(d, n), the maximal number of vertices that a dpolytope with n facets can have according to the Upper Bound Theorem? We show that in dimension d = 4, the a ..."
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of vertices in a monotone path along edges of a ddimensional polytope with n facets can be as large as conceivably possible: Is M(d, n) = Mubt(d, n), the maximal number of vertices that a dpolytope with n facets can have according to the Upper Bound Theorem? We show that in dimension d = 4, the answer is “yes”, despite the fact that it is “no” if we restrict ourselves to the dualtocyclic polytopes. For each n ≥ 5, we exhibit a realization of a polartoneighborly 4dimensional polytope with n facets and a Hamilton path through its vertices that is monotone with respect to a linear objective function. This constrasts an earlier result, by which no polartoneighborly 6dimensional polytope with 9 facets admits a monotone Hamilton path. 1.