Perturbation of transportation polytopes

We describe a perturbation method that can be used to reduce the problem of finding the multivariate generating function (MGF) of a non-simple polytope to computing the MGF of simple polytopes. We then construct a perturbation that works for any transportation polytope. We apply this perturbation to the family of central transportation polytopes of order $kn\times n$, and obtain formulas for the MGFs of the feasible cone of each vertex of the polytope and the MGF of the polytope. The formulas we obtain are enumerated by combinatorial objects. A special case of the formulas recovers the results on Birkhoff polytopes given by the author and De Loera and Yoshida. We also recover the formula for the number of maximum vertices of transportation polytopes of order $kn\times n$.     

A Centrally Symmetric Version of the Cyclic Polytope

We define a centrally symmetric analogue of the cyclic polytope and study its facial structure. We conjecture that our polytopes provide asymptotically the largest number of faces in all dimensions among all centrally symmetric polytopes with n vertices of a given even dimension d=2k when d is fixed and n grows. For a fixed even dimension d=2k and an integer 1≤j<k we prove that the maximum possible number of j-dimensional faces of a centrally symmetric d-dimensional polytope with n vertices is at least for some c _j (d)>0 and at most as n grows. We show that c ₁(d)≥1−(d−1)⁻¹ and conjecture that the bound is best possible. Research of A. Barvinok partially supported by NSF grant DMS 0400617. Research of I. Novik partially supported by Alfred P. Sloan Research Fellowship and NSF grant DMS-0500748.     

Computational results on an algorithm for finding all vertices of a polytope

This paper provides answers to several questions raised by V. Klee regarding the efficacy of Mattheiss' algorithm for finding all vertices of convex polytopes. Several results relating to the expected properties of polytopes are given which indicate thatn-polytopes defined by large numbers of constraints are difficult to obtain by random processes, the expected value of the number of vertices of polytope is considerably less than Klee's least upper bound the expected performance of Mattheiss' algorithm is far better than Klee's upper bound would suggest.     

A criterion for the adjacency of vertices of polytopes generated by subsets of symmetric groups

We propose a characterization of the adjacency of vertices in the class of permutation polytopes generated by arbitrary subsets of symmetric groups. In particular, this class contains polytopes for the well-known classical problems, such as the assignment problem, 2-and 3-combinations, the traveling salesman problem and their various modifications. Up to now, the problem of vertex adjacency has been studied for a single polytope only. In the present paper, we obtain, for general permutation polytopes, necessary and sufficient conditions that guarantee that two given vertices are adjacent (or not) to each other. The conditions are formulated in terms of permutations and of the solvability of certain special systems of linear equations. The presently known adjacency criteria for vertices of polytopes for the assignment problem are simple corollaries of our conditions. The latter allow us to develop a general algorithmic scheme for recognizing vertex adjacency of a general permutation polytope and estimate its complexity.     

An Adjacency Criterion for Coxeter Matroids

A Coxeter matroid is a generalization of matroid, ordinary matroid being the case corresponding to the family of Coxeter groups A _n , which are isomorphic to the symmetric groups. A basic result in the subject is a geometric characterization of Coxeter matroid in terms of the matroid polytope, a result first stated by Gelfand and Serganova. This paper concerns properties of the matroid polytope. In particular, a criterion is given for adjacency of vertices in the matroid polytope.     

Equipartite polytopes

A polytope P with 2n vertices is called equipartite if for any partition of its vertex set into two equal-size sets V ₁ and V ₂, there is an isometry of the polytope P that maps V ₁ onto V ₂. We prove that an equipartite polytope in ℝ ^d can have at most 2d+2 vertices. We show that this bound is sharp and identify all known equipartite polytopes in ℝ ^d . We conjecture that the list is complete.     

Monotonicity of the cd-index for polytopes

We prove that the cd-index of a convex polytope satisfies a strong monotonicity property with respect to the cd-indices of any face and its link. As a consequence, we prove for d-dimensional polytopes a conjecture of Stanley that the cd-index is minimized on the d-dimensional simplex. Moreover, we prove the upper bound theorem for the cd-index, namely that the cd-index of any d-dimensional polytope with n vertices is at most that of C(n,d), the d-dimensional cyclic polytope with n vertices. Received September 29, 1998; in final form February 8, 1999     

On edge-antipodal <Emphasis Type="Italic">d</Emphasis>-polytopes

A convex d-polytope in ℝ ^d is called edge-antipodal if any two vertices that determine an edge of the polytope lie on distinct parallel supporting hyperplanes of the polytope. We introduce a program for investigating such polytopes, and examine those that are simple.      

Upper bounds for edge-antipodal and subequilateral polytopes

A polytope in a finite-dimensional normed space is subequilateral if the length in the norm of each of its edges equals its diameter. Subequilateral polytopes occur in the study of two unrelated subjects: surface energy minimizing cones and edge-antipodal polytopes. We show that the number of vertices of a subequilateral polytope in any d-dimensional normed space is bounded above by (d / 2 + 1) ^d for any d ≥ 2. The same upper bound then follows for the number of vertices of the edge-antipodal polytopes introduced by I. Talata [19]. This is a constructive improvement to the result of A. Pór (to appear) that for each dimension d there exists an upper bound f(d) for the number of vertices of an edge-antipodal d-polytopes. We also show that in d-dimensional Euclidean space the only subequilateral polytopes are equilateral simplices. This material is based upon work supported by the South African National Research Foundation under Grant number 2053752.     

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.     

Facets and volume of Gorenstein Fano polytopes

It is known that every integral convex polytope is unimodularly equivalent to a face of some Gorenstein Fano polytope. It is then reasonable to ask whether every normal polytope is unimodularly equivalent to a face of some normal Gorenstein Fano polytope. In the present paper, it is shown that, by giving new classes of normal Gorenstein Fano polytopes, each order polytope as well as each chain polytope of dimension d is unimodularly equivalent to a facet of some normal Gorenstein Fano polytopes of dimension . Furthermore, investigation on combinatorial properties, especially, Ehrhart polynomials and volume of these new polytopes will be achieved. Finally, some curious examples of Gorenstein Fano polytopes will be discovered.     

New Combinatorial Descriptions of the Triangulations of Cyclic Polytopes and the Second Higher Stasheff–Tamari Posets

This paper is concerned with the d-dimensional cyclic polytope with n vertices, C(n,d), and the set of its triangulations, S(n,d). We show that there is a bijection between S(n,d) and certain partitions of the set of increasing d-tuples on the integers 1 to n–1. We give a combinatorial characterization of the second higher Stasheff–Tamari poset, which is a partial ordering of S(n,d), and we determine its 2-dimension. There is a well-known representation of triangulations of an n-gon by right bracket vectors. We generalize this to cyclic polytopes of higher dimensions.     

McMullen's conditions and some lower bounds for general convex polytopes

We give a lower bound for the number of vertices of a generald-dimensional polytope with a given numberm ofi-faces for eachi = 0,..., d/2 – 1. The tightness of those bounds is proved using McMullen's conditions. Form greater than a small constant, those lower bounds are attained by simpliciali-neighbourly polytopes.     

On the partial order polytope of a digraph

We introduce the partial order polytope of a digraphD, defined as the convex hull of the incidence vectors of all transitive acyclic arc sets ofD. For this polytope we prove some classes of inequalities to be facet-defining and show that there is a polynomial separation algorithm for each of these classes. The results imply a polynomial separation algorithm for a class of valid inequalities of the clique partitioning polytope that includes the two-chorded odd cycle inequalities. The polyhedral results concerning the partial order polytope are of interest since a cutting plane based algorithm to solve the maximum weighted transitive acyclic subdigraph problem can be used to solve the maximum weighted acyclic subdigraph problem, the maximum weighted linear ordering problem and a flexible manufacturing problem. For the acyclic subdigraph polytope we show that the separation of simplet-reinforcedk-fence-inequalities is -complete.     

A model of the coNP-complete non-Hamilton tour decision problem for directed graphs

A truncated permutation matrix polytope is defined as the convex hull of a proper subset of n-permutations represented as 0/1 matrices. We present a linear system that models the coNP-complete non-Hamilton tour decision problem based upon constructing the convex hull of a set of truncated permutation matrix polytopes. Define polytope P^n–1 as the convex hull of all n-1 by n-1 permutation matrices. Each extreme point of P^n–1 is placed in correspondence (a bijection) with each Hamilton tour of a complete directed graph on n vertices. Given any n vertex graph G_n, a polynomial sized linear system F⁽ⁿ⁾ is constructed for which the image of its solution set, under an orthogonal projection, is the convex hull of the complete set of extrema of a subset of truncated permutation matrix polytopes, where each extreme point is in correspondence with each Hamilton tour not in G_n. The non-Hamilton tour decision problem is modeled by F⁽ⁿ⁾ such that G_n is non-Hamiltonian if and only if, under an orthogonal projection, the image of the solution set of F⁽ⁿ⁾ is P^n–1. The decision problem Is the projection of the solution set of F⁽ⁿ⁾=P^n–1? is therefore coNP-complete, and this particular model of the non-Hamilton tour problem appears to be new.Dedicated to the 250+ families in Kelowna BC, who lost their homes due to forest fires in 2003.I visited Ted at his home in Kelowna during this time - his family opened their home to evacuees and we shared happy and sad times with many wonderful people.     

An analog of determinant related to Parshin—Kato theory and integer polytopes

Parshin—Kato theory involves a multilinear function of n+1 vectors in the n-dimensional vector space over the field ?/2?. The same function arises in the computation of the product in the group (?*)ⁿ of all roots of several polynomial equations with sufficiently generic Newton polytopes. We discuss this remarkable function and related geometry of integer polytopes.     

A spectral approach to polyhedral dimension

Thespectrum spec( ) of a convex polytope is defined as the ordered (non-increasing) list of squared singular values of [A|1], where the rows ofA are the extreme points of . The number of non-zeros in spec( ) exceeds the dimension of by one. Hence, the dimension of a polytope can be established by determining its spectrum. Indeed, this provides a new method for establishing the dimension of a polytope, as the spectrum of a polytope can be established without appealing to a direct proof of its dimension. The spectrum is determined for the four families of polytopes defined as the convex hulls of:

1. The edge-incidence vectors of cutsets induced by balanced bipartitions of the vertices in the complete undirected graph on 2_q vertices (see Section 6).
2. The edge-incidence vectors of Hamiltonian tours in the complete undirected graph onn vertices (see Section 6).
3. The arc-incidence vectors of directed Hamiltonian tours in the complete directed graph ofn nodes (see Section 7).
4. The edge-incidence vectors of perfect matchings in the complete 3-uniform hypergraph on 3_q vertices (see Section 8).

In the cases of (ii) and (iii), the associated dimension results are well-known. The dimension results for (i) and (iv) do not seem to be well-known. General principles are discussed for ‘balanced polytopes’ arising from complete structures.     

Momentopes, the Complexity of Vector Partitioning, and Davenport—Schinzel Sequences

The computational complexity of the partition problem , which concerns the partitioning of a set of n vectors in d -space into p parts so as to maximize an objective function which is convex on the sum of vectors in each part, is determined by the number of vertices of the corresponding p-partition polytope defined to be the convex hull in (d\times p) -space of all solutions. In this article, introducing and using the class of Momentopes , we establish the lower bound v _p,d (n)=Ω(n^ \lfloor (d-1)/2 \rfloor p ) on the maximum number of vertices of any p -partition polytope of a set of n points in d -space, which is quite compatible with the recent upper bound v _p,d (n)=O(n ^d(p-1)-1 ) , implying the same bound on the complexity of the partition problem. We also discuss related problems on the realizability of Davenport—Schinzel sequences and describe some further properties of Momentopes. Received April 4, 2001, and in revised form October 3, 2001. Online publication February 28, 2002.     

On the Skeleton of the Polytope of Pyramidal Tours

We consider the skeleton of the polytope of pyramidal tours. A Hamiltonian tour is called pyramidal if the salesperson starts in city 1, then visits some cities in increasing order of their numbers, reaches city n, and returns to city 1 visiting the remaining cities in decreasing order. The polytope PYR(n) is defined as the convex hull of the characteristic vectors of all pyramidal tours in the complete graph K _n . The skeleton of PYR(n) is the graph whose vertex set is the vertex set of PYR(n) and the edge set is the set of geometric edges or one-dimensional faces of PYR(n). We describe the necessary and sufficient condition for the adjacency of vertices of the polytope PYR(n). On this basis we developed an algorithm to check the vertex adjacency with linear complexity. We establish that the diameter of the skeleton of PYR(n) equals 2, and the asymptotically exact estimate of the clique number of the skeleton of PYR(n) is Θ(n²). It is known that this value characterizes the time complexity in a broad class of algorithms based on linear comparisons.