Polytopes with few vertices and few facets

Extension complexity of polytopes with few vertices or facets

We study the extension complexity of polytopes with few vertices or facets. On the one hand, we provide a complete classification of $d$-polytopes with at most $d+4$ vertices according to their extension complexity: Out of the super-exponentially many $d$-polytopes with $d+4$ vertices, all have extension complexity $d+4$ except for some families of size $θ(d^2)$. On the other hand, we show that generic realizations of simplicial/simple $d$-polytopes with $d+1+α$ vertices/facets have extension complexity at least $2 \sqrt{d(d+α)} -d + 1$, which shows that for all $d>(\frac{α-1}{2})^2$ there are…

On hyperbolic Coxeter polytopes with mutually intersecting facets

We prove that, apart from some well-known low-dimensional examples, any compact hyperbolic Coxeter polytope has a pair of disjoint facets. This is one of very few known general results concerning combinatorics of compact hyperbolic Coxeter polytopes. We also obtain a similar result for simple non-compact polytopes.

Facets and facet subgraphs of symmetric edge polytopes

Symmetric edge polytopes, a.k.a. PV-type adjacency polytopes, associated with undirected graphs have been defined and studied in several seemingly independent areas including number theory, discrete geometry, and dynamical systems. In particular, the authors are motivated by the study of the algebraic Kuramoto equations of unmixed form whose Newton polytopes are the symmetric edge polytopes. The interplay between the geometric structure of symmetric edge polytopes and the topological structure of the underlying graphs has been a recurring theme in recent studies. In particular, ``facet/face su…

Long monotone paths on simple 4-polytopes

The Monotone Upper Bound Problem (Klee, 1965) asks if the number M(d,n) of vertices in a monotone path along edges of a d-dimensional polytope with n facets can be as large as conceivably possible: Is M(d,n) = M_{ubt}(d,n), the maximal number of vertices that a d-polytope 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 dual-to-cyclic polytopes. For each n>=5, we exhibit a realization of a polar-to-neighborly 4-dimensional polytope with n facets and a Hamilton pa…

Coxeter polytopes with a unique pair of non-intersecting facets

We consider compact hyperbolic Coxeter polytopes whose Coxeter diagram contains a unique dotted edge. We prove that such a polytope in d-dimensional hyperbolic space has at most d+3 facets. In view of results of Lannér, Kaplinskaja, Esselmann, and the second author, this implies that compact hyperbolic Coxeter polytopes with a unique pair of non-intersecting facets are completely classified. They do exist only up to dimension 6 and in dimension 8.

Near Classification of Compact Hyperbolic Coxeter $d$-Polytopes with $d+4$ Facets and Related Dimension Bounds

We complete the classification of compact hyperbolic Coxeter $d$-polytopes with $d+4$ facets for $d=4$ and $5$. By previous work of Felikson and Tumarkin, the only remaining dimension where new polytopes may arise is $d=6$. We derive a new method for generating the combinatorial type of these polytopes via the classification of point set order types. In dimensions $4$ and $5$, there are $348$ and $51$ polytopes, respectively, yielding many new examples for further study. We furthermore provide new upper bounds on the dimension $d$ of compact hyperbolic Coxeter polytopes with $d+k$ facets for $…

All 2-neighborly d-polytopes with at most d + 9 facets

We give a complete enumeration of all 2-neighborly $d$-polytopes with $d+9$ and less facets. All of them are realized as 0/1-polytopes, except a 6-polytope $P_{6,10,15}$ with 10 vertices and 15 facets, and pyramids over $P_{6,10,15}$. In particular, we update the lower bounds for the number of facets of a 2-neighborly $d$-polytope $P$ and showed that the number of facets of $P$ is not less than the number of its vertices $f_0(P)$ for $f_0(P) \le d + 10$.

Semiregular Polytopes and Amalgamated C-groups

In the classical setting, a convex polytope is said to be semiregular if its facets are regular and its symmetry group is transitive on vertices. This paper studies semiregular abstract polytopes, which have abstract regular facets, still with combinatorial automorphism group transitive on vertices. We analyze the structure of the automorphism group, focusing in particular on polytopes with two kinds of regular facets occurring in an "alternating" fashion. In particular we use group amalgamations to prove that given two compatible n-polytopes P and Q, there exists a universal abstract semiregu…

Reconstructing a Simple Polytope from its Graph

Blind and Mani (1987) proved that the entire combinatorial structure (the vertex-facet incidences) of a simple convex polytope is determined by its abstract graph. Their proof is not constructive. Kalai (1988) found a short, elegant, and algorithmic proof of that result. However, his algorithm has always exponential running time. We show that the problem to reconstruct the vertex-facet incidences of a simple polytope P from its graph can be formulated as a combinatorial optimization problem that is strongly dual to the problem of finding an abstract objective function on P (i.e., a shelling or…

Split-facets for Balanced Minimal Evolution Polytopes and the Permutoassociahedron

Understanding the face structure of the balanced minimal evolution (BME) polytope, especially its top-dimensional facets, is crucially important to phylogenetic applications. We show that BME polytope has a sub-lattice of its poset of faces which is isomorphic to a quotient of the well-studied permutoassociahedron. This sub-lattice corresponds to compatible sets of splits displayed by phylogenetic trees, and extends the lattice of faces of the BME polytope found by Hodge, Haws, and Yoshida. Each of the maximal elements in our new poset of faces corresponds to a single split of the leaves. Near…

Generating facets for the cut polytope of a graph by triangular elimination

The cut polytope of a graph arises in many fields. Although much is known about facets of the cut polytope of the complete graph, very little is known for general graphs. The study of Bell inequalities in quantum information science requires knowledge of the facets of the cut polytope of the complete bipartite graph or, more generally, the complete k-partite graph. Lifting is a central tool to prove certain inequalities are facet inducing for the cut polytope. In this paper we introduce a lifting operation, named triangular elimination, applicable to the cut polytope of a wide range of graphs.…

关于序多面体与链多面体的面间隙

On facet gaps of order and chain polytopes

该研究针对Bhandari等人提出的三个问题，得出有限偏序集的序多面体与链多面体之间面间隙的精确局部公式，还分类了间隙为2的情况并确定标记链-序多面体的面差异。

Vertex-Facet Assignments For Polytopes

Motivated by the search for reduced polytopes, we consider the following question: For which polytopes exists a vertex-facet assignment, that is, a matching between vertices and non-incident facets, so that the matching covers either all vertices, or all facets? We provide general conditions for the existence of such an assignment. We conclude that such exist for all simple and simplicial polytopes, as well as all polytopes of dimension $d\le 6$. We construct counterexample in all dimensions $d\ge 7$.

Tropicalization of facets of polytopes

It is known that any tropical polytope is the image under the valuation map of ordinary polytopes over the Puiseux series field. The latter polytopes are called lifts of the tropical polytope. We prove that any pure tropical polytope is the intersection of the tropical half-spaces given by the images under the valuation map of the facet-defining half-spaces of a certain lift. We construct this lift explicitly, taking into account geometric properties of the given polytope. Moreover, when the generators of the tropical polytope are in general position, we prove that the above property is satisf…

The facets of the matroid polytope and the independent set polytope of a positroid

A positroid is a special case of a realizable matroid that arose from the study of the totally nonnegative part of the Grassmannian by Postnikov. In this paper, we study the facets of its matroid polytope and the independent set polytope. This allows one to describe the bases and independent sets directly from the decorated permutation, bypassing the use of the Grassmann necklace. We also describe a criterion for determining whether a given cyclic interval is a flat or not using the decorated permutation, then show how it applies to checking the concordancy of positroids.

少边图的对称边多胞体的面数

Facets of Symmetric Edge Polytopes for Graphs with Few Edges

本文研究少边连通图的对称边多胞体，引入最大与最小面数序列，建立稀疏图类的面数公式，并推进识别面数最大化图的猜想。

Facets of the Total Matching Polytope for bipartite graphs

The Total Matching Polytope generalizes the Stable Set Polytope and the Matching Polytope. In this paper, we give the perfect formulation for Trees and we derive two new families of valid inequalities, the balanced biclique inequalities which are always facet-defining and the non-balanced lifted biclique inequalities obtained by a lifting procedure, which are facet-defining for bipartite graphs. Finally, we give a complete description for Complete Bipartite Graphs.

两个偏序集多面体的面数之差

Gap between the number of facets of the two poset polytopes

本文研究偏序集的序多面体与链多面体面数之差，利用交叉数给出界的描述，并分类差为一的偏序集。

Facets of the Balanced Minimal Evolution Polytope

A phylogenetic tree is a way to organize a finite set of species, individuals or other sources of related data. The species for which we have existing DNA data make up the set of leaves of the tree. The balanced minimal evolution method of creating phylogenetic trees can be formulated as a linear programming problem, minimizing an inner product over the vertices of the BME polytope. In this paper we undertake the first steps of describing the facets of this polytope.

The lower bound for the number of facets of a k-neighborly d-polytope with d+3 vertices

We have found the minimal difference $Δ(k) = \min\limits_P (f_{d-1}(P) - f_{0}(P))$ between the number of facets and the number of vertices of a $k$-neighborly $d$-polytope $P$ for the case $f_{0}(P) = d+3$: $Δ(2) = 4$, $Δ(3) = 15$, and $Δ(k) = 2 (k^2 - 1)$ for $k \ge 4$.

Primary Facets Of Order Polytopes

Mixture models on order relations play a central role in recent investigations of transitivity in binary choice data. In such a model, the vectors of choice probabilities are the convex combinations of the characteristic vectors of all order relations of a chosen type. The five prominent types of order relations are linear orders, weak orders, semiorders, interval orders and partial orders. For each of them, the problem of finding a complete, workable characterization of the vectors of probabilities is crucial---but it is reputably inaccessible. Under a geometric reformulation, the problem ask…

Volume, facets and dual polytopes of twinned chain polytopes

Let $(P,\leq_P)$ and $(Q,\leq_Q)$ be finite partially ordered sets with $|P|=|Q|=d$, and $\mathcal{C}(P) \subset \mathbb{R}^d$ and $\mathcal{C}(Q) \subset \mathbb{R}^d$ their chain polytopes. The twinned chain polytope of $P$ and $Q$ is the lattice polytope $Γ(\mathcal{C}(P),\mathcal{C}(Q)) \subset \mathbb{R}^d$ which is the convex hull of $\mathcal{C}(P) \cup (-\mathcal{C}(Q))$. It is known that twinned chain polytopes are Gorenstein Fano polytopes with the integer decomposition property. In the present paper, we study combinatorial properties of twinned chain polytopes. First, we will give t…

The facets of the spanning trees polytope

Let $G=(V, E)$ be an undirected graph. The spanning trees polytope $P(G)$ is the convex hull of the characteristic vectors of all spanning trees of $G$. In this paper, we describe all facets of $P(G)$ as a consequence of the facets of the bases polytope $P(M)$ of a matroid $M$, i.e., the convex hull of the characteristic vectors of all bases of $M$.

随机对称边多面体的面、度序列与聚类

Facets of Random Symmetric Edge Polytopes, Degree Sequences, and Clustering

研究随机图模型下对称边多面体的面结构，发现Erdős-Renyi图的阈值概率，并证明固定度序列下平均局部聚类系数与面数量正相关。

非中心对称反射多面体的面数：来自偏序集的双链多面体

Facet numbers of non-centrally symmetric reflexive polytopes arising from posets

本文研究了双链多面体的面数，证明其至多为6^{d/2}，并为Nill猜想提供了部分答案。

Counting vertices of integral polytopes defined by facets

We present a number of complexity results concerning the problem of counting vertices of an integral polytope defined by a system of linear inequalities. The focus is on polytopes with small integer vertices, particularly 0/1 polytopes and half-integral polytopes.

源于联图的对称边多面体的面数

Number of facets of symmetric edge polytopes arising from join graphs

本文研究了连通图对称边多面体面数的上下界猜想，证明了该猜想对任意两图的联图成立，从而为 Nill 关于反身多面体面数上界的猜想提供了部分回答。

Weighted graphs defining facets: a connection between stable set and linear ordering polytopes

A graph is alpha-critical if its stability number increases whenever an edge is removed from its edge set. The class of alpha-critical graphs has several nice structural properties, most of them related to their defect which is the number of vertices minus two times the stability number. In particular, a remarkable result of Lovász (1978) is the finite basis theorem for alpha-critical graphs of a fixed defect. The class of alpha-critical graphs is also of interest for at least two topics of polyhedral studies. First, Chvátal (1975) shows that each alpha-critical graph induces a rank inequality…

An Efficient Approximation of the Traveling Salesman Polytope Using Lifting Methods

For the Traveling Salesman Polytope on n cities T_n, we construct its approximation Q_k, k=1, 2, . . ., n^(1/3) using a projection of a polytope whose number of facets is polynomial in n (of degree linear in k). We show that T_n is contained in Q_k for each k, and that the scaling of Q_k by k/n+O(1/n) is contained in T_n for each k. We show that certain facets of T_n lie on the boundary of Q_k.

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 $d + 1$. Furthermore, investigation on combinatorial properties, especially, Ehrhart …

Facets of the (s,t)-p-path polytope

We give a partial description of the (s,t)-p-path polytope of a directed graph D which is the convex hull of the incidence vectors of simple directed (s,t)-paths in D of length p. First, we point out how the (s,t)-p-path polytope is located in the family of path and cycle polyhedra. Next, we give some classes of valid inequalities which are very similar to inequalities which are valid for the p-cycle polytope, that is, the convex hull of the incidence vectors of simple cycles of length p in D. We give necessary and sufficient conditions for these inequalities to be facet defining. Furthermore,…

凸多面体的面枚举算法

A Facet Enumeration Algorithm for Convex Polytopes

本文提出一种简单新颖的凸多面体面枚举算法，在Matlab中实现，并通过已知多面体测试验证其有效性和效率，且因对偶性可推广至顶点枚举。

A Positive Semidefinite Approximation of the Symmetric Traveling Salesman Polytope

For a convex body B in a vector space V, we construct its approximation P_k, k=1, 2, . . . using an intersection of a cone of positive semidefinite quadratic forms with an affine subspace. We show that P_k is contained in B for each k. When B is the Symmetric Traveling Salesman Polytope on n cities T_n, we show that the scaling of P_k by n/k+ O(1/n) contains T_n for k no more than n/2. Membership for P_k is computable in time polynomial in n (of degree linear in k). We discuss facets of T_n that lie on the boundary of P_k. We introduce a new measure on each facet defining inequality for T_n in…

Flat extensions of abstract polytopes

We consider the problem of constructing an abstract $(n+1)$-polytope $Q$ with $k$ facets isomorphic to a given $n$-polytope $P$, where $k \geq 3$. In particular, we consider the case where we want $Q$ to be $(n-2,n)$-flat, meaning that every $(n-2)$-face is incident to every $n$-face (facet). We show that if $P$ admits such a flat extension for a given $k$, then the facet graph of $P$ is $(k-1)$-colorable. Conversely, we show that if the facet graph is $(k-1)$-colorable and $k-1$ is prime, then $P$ admits a flat extension for that $k$. We also show that if $P$ is facet-bipartite, then for ever…

半正抽象多面体及其平凡面稳定化子

Semiregular abstract polyhedra with trivial facet stabilizer

本文构造了具有平凡面稳定化子的半正抽象多面体，展示了其可拥有无界的旗轨道数，同时仅有一个面轨道，并通过操作高阶正则和非正则多面体实现交替半正多面体的构造。

Directed random walks on polytopes with few facets

Let $P$ be a simple polytope with $n-d = 2$, where $d$ is the dimension and $n$ is the number of facets. The graph of such a polytope is also called a grid. It is known that the directed random walk along the edges of $P$ terminates after $O(\log^2 n)$ steps, if the edges are oriented in a (pseudo-)linear fashion. We prove that the same bound holds for the more general unique sink orientations.

Polyhedral aspects of score equivalence in Bayesian network structure learning

This paper deals with faces and facets of the family-variable polytope and the characteristic-imset polytope, which are special polytopes used in integer linear programming approaches to statistically learn Bayesian network structure. A common form of linear objectives to be maximized in this area leads to the concept of score equivalence (SE), both for linear objectives and for faces of the family-variable polytope. We characterize the linear space of SE objectives and establish a one-to-one correspondence between SE faces of the family-variable polytope, the faces of the characteristic-imset…

Derivations of large classes of facet-defining inequalities of the weak order polytope using ranking structures

The study of ordering polytopes has been essential to the solution of various challenging combinatorial optimization problems. For instance, the incorporation of facet defining inequalities (FDIs) from these polytopes in branch-and-cut approaches represents among the most effective solution methodologies known to date for some of these problems. The weak order polytope, defined as the convex hull of the characteristic vectors of all binary orders on $n$ alternatives that are reflexive, transitive, and complete, has been particularly important for tackling problems in computational social choic…

Facet volumes of polytopes

In this paper, motivated by the work of Edelman and Strang, we show that for fixed integers $d\geq 2$ and $n\geq d+1$ the configuration space of all facet volume vectors of all $d$-polytopes in $\mathbb R^{d}$ with $n$ facets is a full dimensional cone in $\mathbb R^{n}$. In particular, for tetrahedra ($d=3$ and $n=4$) this is a cone over a regular octahedron. Our proof is based on a novel configuration space / test map scheme which uses topological methods for finding solutions of a problem, and tools of differential geometry to identify solutions with the desired properties. Furthermore, our…

Face numbers of 4-Polytopes and 3-Spheres

In this paper, we discuss f- and flag-vectors of 4-dimensional convex polytopes and cellular 3-spheres. We put forward two crucial parameters of fatness and complexity: Fatness F(P) := (f_1+f_2-20)/(f_0+f_3-10) is large if there are many more edges and 2-faces than there are vertices and facets, while complexity C(P) := (f_{03}-20)/(f_0+f_3-10) is large if every facet has many vertices, and every vertex is in many facets. Recent results suggest that these parameters might allow one to differentiate between the cones of f- or flag-vectors of -- connected Eulerian lattices of length 5 (combinato…

Non-flat regular polytopes and restrictions on chiral polytopes

An abstract polytope is \emph{flat} if every facet is incident on every vertex. In this paper, we prove that no chiral polytope has flat finite regular facets and finite regular vertex-figures. We then determine the three smallest non-flat regular polytopes in each rank, and use this to show that for $n \geq 8$, a chiral $n$-polytope has at least $48(n-2)(n-2)!$ flags.

0/1多面体的面数的无对数下界

A Log-Free Lower Bound for the Number of Facets of $0/1$-Polytopes

该研究针对0/1多面体，移除了前人下界中的对数因子，通过随机符号多面体与拉德马赫速率体的比较等方法，得到了更优的面数下界。

The Birkhoff polytope of the groups $\mathsf{F}_4$ and $\mathsf{H}_4$

We compute the set of facets of the polytope which is the convex hull of the Coxeter groups $\mathsf{F}_4$ or $\mathsf{H}_4$: For the group $\mathsf{F}_4$ we found $2$ orbits of facets which contradicts previous results published in \cite{birkhoff}. For the group $\mathsf{H}_4$ we found $1063$ orbits of facets which provides a counterexample to the conjecture of \cite{birkhoff}.

Computational methods for cones and polytopes with symmetry

Every polyhedral cone can be described either by its facets or by its extreme rays. Computation of one description from the other is a problem that can be very complex, i.e. one encounter the combinatorial explosion. We present here several methods which allow us to use symmetries for the computation of extreme rays and facets of cones. I give an algorithm for computing adjacencies using a linear programming approach. I give a complete description of the method of incidence, adjacency decomposition method and of the subcone method used for computing with symmetric cones.

Chiral extensions of chiral polytopes

Given a chiral d-polytope K with regular facets, we describe a construction for a chiral (d + 1)-polytope P with facets isomorphic to K. Furthermore, P is finite whenever K is finite. We provide explicit examples of chiral 4-polytopes constructed in this way from chiral toroidal maps.

具有10个面的紧双曲考克斯特六维多面体

Compact Hyperbolic Coxeter Six-dimensional Polytopes With Ten Facets

该研究完成了所有维度下具有$d+4$个面的紧双曲考克斯特$d$维多面体的分类，证明了具有10个面的紧双曲考克斯特6维多面体唯一，并通过精确方法完成了候选类型的穷举验证。

A bound for the number of vertices of a polytope with applications

We prove that the number of vertices of a polytope of a particular kind is exponentially large in the dimension of the polytope. As a corollary, we prove that an n-dimensional centrally symmetric polytope with O(n) facets has 2^{Omega(n)} vertices and that the number of r-factors in a k-regular graph is exponentially large in the number of vertices of the graph provided k >2r and every cut in the graph with at least two vertices on each side has more than k/r edges.

The contact polytope of the Leech lattice (complete version)

The contact polytope of a lattice is the convex hull of its shortest vectors. In this paper we classify the facets of the contact polytope of the Leech lattice up to symmetry. There are 1,197,362,269,604,214,277,200 many facets in 232 orbits.

小格点多面体具有较少的顶点

Small lattice polytopes have few vertices

通过几何化Konyagin和Sevast'yanov对Andrews定理的证明，给出格点多面体顶点数关于体积的紧上界。

关于表示比较概率序的多面体的面数研究

On the Number of Facets of Polytopes Representing Comparative Probability Orders

针对比较概率序几何表示中区域闭包多面体的最大面数问题，研究采用组合方法结合可翻转对概念，证明最大面数下界为第n+1个斐波那契数且推测其为紧界，同时得到差距不大的上界。

自反多胞体的Oda猜想：若干特殊情形

Oda's conjecture for reflexive polytopes: some special cases

本文在若干特殊条件下证明了自反多胞体的Oda猜想，包括单纯自反多胞体与含原点的格多胞体满足特定顶点和格点条件的情形，以及面幺模多胞体和几乎余幺模自反多胞体对的情形。

The Hamiltonian Circuit Polytope

The hamiltonian circuit polytope is the convex hull of feasible solutions for the circuit constraint, which provides a succinct formulation of the traveling salesman and other sequencing problems. We study the polytope by establishing its dimension, developing tools for the identification of facets, and using these tools to derive several families of facets. The tools include necessary and sufficient conditions for an inequality to be facet defining, and an algorithm for generating all undominated circuits. We use a novel approach to identifying families of facet-defining inequalities, based o…

New Classes of Facets of Cut Polytope and Tightness of I_{mm22} Bell Inequalities

The Grishukhin inequality Gr_7 is a facet of CutP_7, the cut polytope on seven points, which is ``sporadic'' in the sense that its proper generalization has not been known. In this paper, we extend Gr_7 to an inequality I(G,H) valid for CutP_{n+1} where G and H are graphs with n nodes satisfying certain conditions, and prove a necessary and sufficient condition for I(G,H) to be a facet. This result combined with the triangular elimination theorem of Avis, Imai, Ito and Sasaki settles Collins and Gisin's conjecture in quantum theory affirmatively: the I_{mm22} Bell inequality is a facet of the …

The stable set polytope of ($P_6$,triangle)-free graphs and new facet-inducing graphs

The stable set polytope of a graph $G$, denoted as STAB($G$), is the convex hull of all the incidence vectors of stable sets of $G$. To describe a linear system which defines STAB($G$) seems to be a difficult task in the general case. In this paper we present a complete description of the stable set polytope of ($P_6$,triangle)-free graphs (and more generally of ($P_6$,paw)-free graphs). For that we combine different tools, in the context of a well known result of Chvátal \cite{Chvatal1975} which allows to focus just on prime facet-inducing graphs, with particular reference to a structure resu…