A Continuous <Emphasis Type="Italic">d</Emphasis>-Step Conjecture for Polytopes

The curvature of a polytope, defined as the largest possible total curvature of the associated central path, can be regarded as a continuous analogue of its diameter. We prove an analogue of the result of Klee and Walkup. Namely, we show that if the order of the curvature is less than the dimension d for all polytopes defined by 2d inequalities and for all d, then the order of the curvature is less that the number of inequalities for all polytopes.     

On the set covering polytope: II. Lifting the facets with coefficients in {0, 1, 2}

In an earlier paper (Mathematical Programming 43 (1989) 57–69) we characterized the class of facets of the set covering polytope defined by inequalities with coefficients equal to 0, 1 or 2. In this paper we connect that characterization to the theory of facet lifting. In particular, we introduce a family of lower dimensional polytopes and associated inequalities having only three nonzero coefficients, whose lifting yields all the valid inequalities in the above class, with the lifting coefficients given by closed form expressions.The research underlying this report was supported by Grant ECS-8601660 of the National Science Foundation, Contract N00014-85-K-0198 with the Office of Naval Research, and Grant AFOSR-870292 of the Air Force Office of Scientific Research.     

Box-inequalities for quadratic assignment polytopes

Linear Programming based lower bounds have been considered both for the general as well as for the symmetric quadratic assignment problem several times in the recent years. Their quality has turned out to be quite good in practice. Investigations of the polytopes underlying the corresponding integer linear programming formulations (the non-symmetric and the symmetric quadratic assignment polytope) have been started during the last decade [34, 31, 21, 22]. They have lead to basic knowledge on these polytopes concerning questions like their dimensions, affine hulls, and trivial facets. However, no large class of (facet-defining) inequalities that could be used in cutting plane procedures had been found. We present in this paper the first such class of inequalities, the box inequalities, which have an interesting origin in some well-known hypermetric inequalities for the cut polytope. Computational experiments with a cutting plane algorithm based on these inequalities show that they are very useful with respect to the goal of solving quadratic assignment problems to optimality or to compute tight lower bounds. The most effective ones among the new inequalities turn out to be indeed facet-defining for both the non-symmetric as well as for the symmetric quadratic assignment polytope. Received: April 17, 2000 / Accepted: July 3, 2001?Published online September 3, 2001     

Piles of Cubes, Monotone Path Polytopes, and Hyperplane Arrangements

Monotone path polytopes arise as a special case of the construction of fiber polytopes, introduced by Billera and Sturmfels. A simple example is provided by the permutahedron, which is a monotone path polytope of the standard unit cube. The permutahedron is the zonotope polar to the braid arrangement. We show how the zonotopes polar to the cones of certain deformations of the braid arrangement can be realized as monotone path polytopes. The construction is an extension of that of the permutahedron and yields interesting connections between enumerative combinatorics of hyperplane arrangements and geometry of monotone path polytopes. Received January 24, 1997, and in revised form April 8, 1997.     

Long monotone paths on simple 4-polytopes

TheMonotone Upper Bound Problem (Klee, 1965) asks if the maximal numberM(d,n) of vertices in a monotone path along edges of ad-dimensional polytope withn facets can be as large as conceivably possible: IsM(d,n)=M _ubt (d,n), the maximal number of vertices that ad-polytope withn facets can have according to the Upper Bound Theorem?We show that in dimensiond=4, the answer is “yes”, despite the fact that it is “no” if we restrict ourselves to the dual-to-cyclic polytopes. For eachn≥5, we exhibit a realization of a polar-to-neighborly 4-dimensional polytope withn 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 polar-to-neighborly 6-dimensional polytope with 9 facets admits a monotone Hamilton path.     

Binary positive semidefinite matrices and associated integer polytopes

We consider the positive semidefinite (psd) matrices with binary entries, along with the corresponding integer polytopes. We begin by establishing some basic properties of these matrices and polytopes. Then, we show that several families of integer polytopes in the literature—the cut, boolean quadric, multicut and clique partitioning polytopes—are faces of binary psd polytopes. Finally, we present some implications of these polyhedral relationships. In particular, we answer an open question in the literature on the max-cut problem, by showing that the rounded psd inequalities define a polytope.     

Facial structures of lattice path matroid polytopes

A lattice path matroid is a transversal matroid corresponding to a pair of lattice paths on the plane. A matroid base polytope is the polytope whose vertices are the incidence vectors of the bases of the given matroid. In this paper, we study the facial structures of matroid base polytopes corresponding to lattice path matroids. In the case of a border strip, we show that all faces of a lattice path matroid polytope can be described by certain subsets of deletions, contractions, and direct sums. In particular, we express them in terms of the lattice path obtained from the border strip. Subsequently, the facial structures of a lattice path matroid polytope for a general case are explained in terms of certain tilings of skew shapes inside the given region.     

Minimal Ellipsoid Circumscribing a Polytope Defined by a System of Linear Inequalities

In this paper, we will propose algorithms for calculating a minimal ellipsoid circumscribing a polytope defined by a system of linear inequalities. If we know all vertices of the polytope and its cardinality is not very large, we can solve the problem in an efficient manner by a number of existent algorithms. However, when the polytope is defined by linear inequalities, these algorithms may not work since the cardinality of vertices may be huge. Based on a fact that vertices determining an ellipsoid are only a fraction of these vertices, we propose algorithms which iteratively calculate an ellipsoid which covers a subset of vertices. Numerical experiment shows that these algorithms perform well for polytopes of dimension up to seven.     

Complexity analysis of the analytic center cutting plane method that uses multiple cuts

We analyze the complexity of the analytic center cutting plane or column generation algorithm for solving general convex problems defined by a separation oracle. The oracle is called at the analytic center of a polytope, which contains a solution set and is given by the intersection of the linear inequalities previously generated from the oracle. If the center is not in the solution set, separating hyperplanes will be placed through the center to shrink the containing polytope. While the complexity result has been recently established for the algorithm when one cutting plane is placed in each iteration, the result remains open when multiple cuts are added. Moreover, adding multiple cuts actually is a key to practical effectiveness in solving many problems and it presents theoretical difficulties in analyzing cutting plane methods. In this paper, we show that the analytic center cutting plane algorithm, with multiple cuts added in each iteration, still is a fully polynomial approximation algorithm. The research of the author is supported by NSF grant DDM-9207347, an Iowa Business School Summer Grant, and a University of Iowa Obermann Fellowship.     

On the $${\mathcal {}}$$-free extension complexity of the TSP

It is known that the extension complexity of the TSP polytope for the complete graph \(K_n\) is exponential in n even if the subtour inequalities are excluded. In this article we study the polytopes formed by removing other subsets \({\mathcal {H}}\) of facet-defining inequalities of the TSP polytope. In particular, we consider the case when \({\mathcal {H}}\) is either the set of blossom inequalities or the simple comb inequalities. These inequalities are routinely used in cutting plane algorithms for the TSP. We show that the extension complexity remains exponential even if we exclude these inequalities. In addition we show that the extension complexity of polytope formed by all comb inequalities is exponential. For our proofs, we introduce a subclass of comb inequalities, called (h, t)-uniform inequalities, which may be of independent interest.     

Maintaining closeness to the analytic center of a polytope by perturbing added hyperplanes

In this work we consider a regionR in ℝ ⁿ given by a finite number of linear inequalities and having nonempty interior. We assume a pointx ^o is given, which is close in certain norm to the analytic center ofR, and that a new linear inequality is added to those definingR. It is constructively shown how to obtain a perturbation of the right-hand side of this inequality such that the pointx ^o is still close, in the same norm, to the analytic center of this perturbed polytope. This fact plays a central role in interior point postoptimality techniques for linear programming involving methods of centers.     

Gorenstein polytopes and their stringy E-functions

Inspired by ideas from algebraic geometry, Batyrev and the first named author have introduced the stringy E-function of a Gorenstein polytope. We prove that this a priori rational function is actually a polynomial, which is part of a conjecture of Batyrev and the first named author. The proof relies on a comparison result for the lattice point structure of a Gorenstein polytope P, a face F of P and the face of the dual Gorenstein polytope corresponding to F. In addition, we study joins of Gorenstein polytopes and introduce the notion of an irreducible Gorenstein polytope. We show how these concepts relate to the decomposition of nef-partitions.     

Flag enumerations of matroid base polytopes

In this paper, we study flag structures of matroid base polytopes. We describe faces of matroid base polytopes in terms of matroid data, and give conditions for hyperplane splits of matroid base polytopes. Also, we show how the cd-index of a polytope can be expressed when a polytope is split by a hyperplane, and apply these to the cd-index of a matroid base polytope of a rank 2 matroid.     

Faces for a linear inequality in 0–1 variables

Given a linear inequality in 0–1 variables we attempt to obtain the faces of the integer hull of 0–1 feasible solutions. For the given inequality we specify how faces of a variety of lower-dimensional inequalities can be raised to give full-dimensional faces. In terms of a set, called a “strong cover”, we obtain necessary and sufficient conditions for any inequality with 0–1 coefficients to be a face, and characterize different forms that the integer hull must take. In general the suggested procedures fail to produce the complete integer hull. Special subclasses of inequalities for which all faces can be generated are demonstrated. These include the “matroidal” and “graphic” inequalities, where a count on the number of such inequalities is obtained, and inequalities where all faces can be derived from lower dimensional faces.     

On Facets of Knapsack Equality Polytopes

The 0/1 knapsack equality polytope is, by definition, the convex hull of 0/1 solutions of a single linear equation. A special form of this polytope, where the defining linear equation has nonnegative integer coefficients and the number of variables having coefficient one exceeds the right-hand side, is considered. Equality constraints of this form arose in a real-world application of integer programming to a truck dispatching scheduling problem. Families of facet defining inequalities for this polytope are identified, and complete linear inequality representations are obtained for some classes of polytopes.     

The existence of non-tiles and non-facets in three dimensions

Non-tiles are convex polytopes, of which isomorphic copies will not tile the space locally finite and face-to-face (that is, neighbouring tiles meet in a face of each). Non-facets are convex polytopes, of which isomorphic copies will not fit together as the facets of a higher dimensional convex polytope. It is proved that there are infinitely many 3-dimensional convex polytopes which are non-tiles as well as non-facets, thereby answering a question of Danzer.     

Polyhedral results for the edge capacity polytope

Network loading problems occur in the design of telecommunication networks, in many different settings. For instance, bifurcated or non-bifurcated routing (also called splittable and unsplittable) can be considered. In most settings, the same polyhedral structures return. A better understanding of these structures therefore can have a major impact on the tractability of polyhedral-guided solution methods. In this paper, we investigate the polytopes of the problem restricted to one arc/edge of the network (the undirected/directed edge capacity problem) for the non-bifurcated routing case.?As an example, one of the basic variants of network loading is described, including an integer linear programming formulation. As the edge capacity problems are relaxations of this network loading problem, their polytopes are intimately related. We give conditions under which the inequalities of the edge capacity polytopes define facets of the network loading polytope. We describe classes of strong valid inequalities for the edge capacity polytopes, and we derive conditions under which these constraints define facets. For the diverse classes the complexity of lifting projected variables is stated.?The derived inequalities are tested on (i) the edge capacity problem itself and (ii) the described variant of the network loading problem. The results show that the inequalities substantially reduce the number of nodes needed in a branch-and-cut approach. Moreover, they show the importance of the edge subproblem for solving network loading problems. Received: September 2000 / Accepted: October 2001?Published online March 27, 2002