We consider the {\em Shaped Partition Problem} of partitioning $n$ given vectors in real $k$-space into $p$ parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part, subject to arbitrary constraints on the number of elements in each part. In addressing this problem, we study the {\em Shaped Partition Polytope} defined as the convex hull of solutions. The Shaped Partition Problem captures ${\cal N}{\cal P}$-hard problems such as the Max-Cut problem and the Traveling Salesperson problem, and the Shaped Partition Polytope may have exponentially many vertices and facets, even when $k$ or $p$ are fixed. In contrast, we show that when both $k$ and $p$ are fixed, the number of vertices is polynomial in $n$, and all vertices can be enumerated and the optimization problem solved in strongly polynomial time. Explicitly, we show that any Shaped Partition Polytope has $O(n^{k{p\choose 2}})$ vertices which can be enumerated in $O(n^{k^2p^3})$ arithmetic operations, and that any Shaped Partition Problem is solvable in $O(n^{kp^2})$ arithmetic operations.
In this paper we investigate the Ehrhart Theory of the independence matroid polytope of uniform matroids. It is proved that these polytopes have an Ehrhart polynomial with positive coefficients. To do that, we prove that indeed all half-open-hypersimplices are Ehrhart positive, and tile disjointly our polytope using them.
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.
We describe a provably complete algorithm for the generation of a tight, possibly exact superset of all combinatorially distinct simple n-facet polytopes in R^d, along with their graphs, f-vectors, and face lattices. The technique applies repeated cutting planes and planar sweeps to a d-simplex. Our generator has implications for several outstanding problems in polytope theory, including conjectures about the number of distinct polytopes, the edge expansion of polytopal graphs, and the d-step conjecture.
We develop a procedure for the complete computational enumeration of lattice $3$-polytopes of width larger than one, up to any given number of lattice points. We also implement an algorithm for doing this and enumerate those with at most eleven lattice points (there are 216,453 of them). In order to achieve this we prove that if $P$ is a lattice 3-polytope of width larger than one and with at least seven lattice points then it fits in one of three categories that we call boxed, spiked and merged. Boxed polytopes have at most 11 lattice points; in particular they are finitely many, and we enume…
A well known result by Lagarias and Ziegler states that there are finitely many equivalence classes of d-dimensional lattice polytopes having volume at most K, for fixed constants d and K. We describe an algorithm for the complete enumeration of such equivalence classes for arbitrary constants d and K. The algorithm, which gives another proof of the finiteness result, is implemented for small values of K, up to dimension six. The resulting database contains and extends several existing ones, and has been used to correct mistakes in other classifications. When specialized to three-dimensional s…
We describe an algorithm to enumerate polytopes. This algorithm is then implemented to give a complete classification of combinatorial spheres of dimension 3 with 9 vertices and decide polytopality of those spheres. In particular, we completely enumerate all combinatorial types of 4-dimensional polytopes with 9 vertices. It is shown that all of those combinatorial types are rational: They can be realized with rational coordinates. We find 316014 combinatorial spheres on 9 vertices. Of those, 274148 can be realized as the boundary complex of a four-dimensional polytope and the remaining 41866 a…
A (convex) polytope $P$ is said to be $2$-level if for every direction of hyperplanes which is facet-defining for $P$, the vertices of $P$ can be covered with two hyperplanes of that direction. The study of these polytopes is motivated by questions in combinatorial optimization and communication complexity, among others. In this paper, we present the first algorithm for enumerating all combinatorial types of $2$-level polytopes of a given dimension $d$, and provide complete experimental results for $d \leqslant 7$. Our approach is inductive: for each fixed $(d-1)$-dimensional $2$-level polytop…
We report here a computation giving the complete list of facets for the cut polytopes over several very symmetric graphs with $15-30$ edges, including $K_8$, $K_{3,3,3}$, $K_{1,4,4}$, $K_{5,5}$, some other $K_{l,m}$, $K_{1,l,m}$, $Prism_7, APrism_6$, Möbius ladder $M_{14}$, Dodecahedron, Heawood and Petersen graphs. For $K_8$, it shows that the huge lists of facets of the cut polytope $CUTP_8$ and cut cone $CUT_8$, given in [CR] is complete. We also confirm the conjecture that any facet of $CUTP_8$ is adjacent to a triangle facet. The lists of facets for $K_{1,l,m}$ with $(l,m)=(4,4),(3,5),(3,…
Enumeration of all combinatorial types of point configurations and polytopes is a fundamental problem in combinatorial geometry. Although many studies have been done, most of them are for 2-dimensional and non-degenerate cases. Finschi and Fukuda (2001) published the first database of oriented matroids including degenerate (i.e. non-uniform) ones and of higher ranks. In this paper, we investigate algorithmic ways to classify them in terms of realizability, although the underlying decision problem of realizability checking is NP-hard. As an application, we determine all possible combinatorial t…
Over the last years the vertex enumeration problem of polyhedra has seen a revival in the study of metabolic networks, which increased the demand for efficient vertex enumeration algorithms for high-dimensional polyhedra given by inequalities. It is a famous and long standing open question in polyhedral theory and computational geometry whether the vertices of a polytope (bounded polyhedron), described by a set of linear constraints, can be enumerated in total polynomial time. In this paper we apply the concept of branch-decomposition to the vertex enumeration problem of polyhedra $P = \{x : A…
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…
The combinatorics of reduced words and commutation classes plays an important role in geometric representation theory. A string polytope is a lattice polytope associated to each reduced word of the longest element $w_0$ in the symmetric group which encodes the character of a certain irreducible representation of a Lie group of type $A$. In this paper, we provide a recursive formula for the number of reduced words of $w_0$ such that the corresponding string polytopes are combinatorially equivalent to a Gelfand-Cetlin polytope. The recursive formula involves the number of standard Young tableaux…
We give a complete enumeration of all 2-neighborly 0/1-polytopes of dimension 7. There are 13 959 358 918 different 0/1-equivalence classes of such polytopes. They form 5 850 402 014 combinatorial classes and 1 274 089 different f-vectors. It enables us to list some of their combinatorial properties. In particular, we have found a 2-neighborly polytope with 14 vertices and 16 facets.
We introduce a natural weighted enumeration of lattice points in a polytope, and give a Brion-type formula for the corresponding generating function. The weighting has combinatorial significance, and its generating function may be viewed as a generalization of the Rogers-Szegő polynomials. It also arises from the geometry of the toric arc scheme associated to the normal fan of the polytope. We show that the asymptotic behavior of the coefficients at $q = 1$ is Gaussian.
It is proved that a certain symmetric sequence of nonnegative integers arising in the enumeration of magic squares of given size n by row sums or, equivalently, in the generating function of the Ehrhart polynomial of the polytope of doubly stochastic n by n matrices, is equal to the h-vector of a simplicial polytope and hence that it satisfies the conditions of the g-theorem. The unimodality of this sequance, which follows, was conjectured by Stanley (1983). Several generalizations are given.
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 $f_0(P) \le d + 10$.
This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and hypergraphs, relational structures, and others. The second part advertises five topics in general enumeration: 1. counting lattice points in lattice polytopes, 2. growth of context-free languages, 3. holonomicity (i.e., P-recursiveness) of numbers of labeled regular graphs, 4. frequent occurrence of the asymptotics cn^{-3/2}r^n and 5. ultimate modular periodicit…
Let $Q_n$ denote the $n$-dimensional hypercube with the vertex set $V_n=\{0,1}^n$. A 0/1-polytope of $Q_n$ is a convex hull of a subset of $V_n$. This paper is concerned with the enumeration of equivalence classes of full-dimensional 0/1-polytopes under the symmetries of the hypercube. With the aid of a computer program, Aichholzer completed the enumeration of equivalence classes of full-dimensional 0/1-polytopes for $Q_4$, $Q_5$, and those of $Q_6$ up to 12 vertices. In this paper, we present a method to compute the number of equivalence classes of full-dimensional 0/1-polytopes of $Q_n$ with…
We consider the semigroup S of highest weights appearing in tensor powers V^k of a finite dimensional representation V of a connected reductive group. We describe the cone generated by S as the cone over the weight polytope of V intersected with the positive Weyl chamber. From this we get a description for the asymptotic of the number of highest weights appearing in V^k in terms of the volume of this polytope.
A \emph{magic square} is an $n \times n$ array of distinct positive integers whose sum along any row, column, or main diagonal is the same number. We compute the number of such squares for $n=4$, as a function of either the magic sum or an upper bound on the entries. The previous record for both functions was the $n=3$ case. Our methods are based on inside-out polytopes, i.e., the combination of hyperplane arrangements and Ehrhart's theory of lattice-point enumeration.
We completely solve the problem of enumerating combinatorially inequivalent $d$-dimensional polytopes with $d+3$ vertices. A first solution of this problem, by Lloyd, was published in 1970. But the obtained counting formula was not correct, as pointed out in the new edition of Grünbaum's book. We both correct the mistake of Lloyd and propose a more detailed and self-contained solution, relying on similar preliminaries but using then a different enumeration method involving automata. In addition, we introduce and solve the problem of counting oriented and achiral (i.e. stable under reflection) …
The coefficients of the chain polynomial of a finite poset enumerate chains in the poset by their number of elements. The chain polynomials of the partition lattices and their standard type $B$ analogues are shown to have only real roots. The real-rootedness of the chain polynomial is conjectured for all geometric lattices and is shown to be preserved by the pyramid and the prism operations on Cohen--Macaulay posets. As a result, new families of convex polytopes whose face lattices have real-rooted chain polynomials are presented. An application to the face enumeration of the second barycentri…
We augment the list of finite universal locally toroidal regular polytopes of type {3,3,4,3,3} due to P.McMullen and E.Schulte, adding as well as removing entries. This disproves a related long-standing conjecture. Our new universal polytope is related to a well-known Y-shaped presentation for the sporadic simple group $Fi_{22}$, and admits $S_4\times O_8^+(2){:}S_3$ as the automorphism group. We also discuss further extensions of its quotients in the context of Y-shaped presentations. As well, we note that two known examples of finite universal polytopes of type {3,3,4,3,3} are related to Y-s…
Let $f(x_1,...,x_k)$ be a polynomial over a field $K$. This paper considers such questions as the enumeration of the number of nonzero coefficients of $f$ or of the number of coefficients equal to $α\in K^*$. For instance, if $K=\ff_q$ then a matrix formula is obtained for the number of coefficients of $f^n$ that are equal to $α\in \ff_q^*$, as a function of $n$. Many additional results are obtained related to such areas as lattice path enumeration and the enumeration of integer points in convex polytopes.
The flag vector contains all the face incidence data of a polytope, and in the poset setting, the chain enumerative data. It is a classical result due to Bayer and Klapper that for face lattices of polytopes, and more generally, Eulerian graded posets, the flag vector can be written as a cd-index, a non-commutative polynomial which removes all the linear redundancies among the flag vector entries. This result holds for regular CW complexes. We relax the regularity condition to show the cd-index exists for Whitney stratified manifolds by extending the notion of a graded poset to that of a quasi…
We use the notions of reflexivity and of reflexive dimensions in order to introduce probability measures for lattice polytopes and initiate the investigation of their statistical properties. Examples of applications to discrete geometry include a study of randomness of self-duality of reflexive polytopes and implications for expectation values of the numbers of such polytopes in higher dimensions. We also discuss enumeration problems and related algorithms and point out interesting open problems. In this context we define the notion of IP-confined polytopes. Our new results include the list of…
A Counterexample to Ziegler's Cross-Polytope Conjecture for Simplicial 0/1-Polytopes
Faculty of Mathematics, Otto-von-Guericke-Universität Magdeburg（马格德堡奥托·冯·古里克大学数学学院）；Institute of Mathematics, Technische Universität Berlin（柏林工业大学数学研究所）；Zuse Institute Berlin（柏林祖斯研究所）
The $k$-tiling problem for a convex polytope $P$ is the problem of covering $\mathbb R^d$ with translates of $P$ using a discrete multiset $Λ$ of translation vectors, such that every point in $\mathbb R^d$ is covered exactly $k$ times, except possibly for the boundary of $P$ and its translates. A classical result in the study of tiling problems is a theorem of McMullen that a convex polytope $P$ that 1-tiles $\mathbb R^d$ with a discrete multiset $Λ$ can, in fact, 1-tile $\mathbb R^d$ with a lattice $\mathcal L$. A generalization of McMullen's theorem for $k$-tiling was conjectured by Gravin, …
The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but non-linear and non-convex constraint. The main results of this paper are significant improvements upon the best known asymptotic bounds on the number of admissible vertices, using polytopes in both the standard normal surface coordinate system and the streamlined quadrilateral coordinate system. To achieve these results we examine the layout of admissible point…
We present an algorithm for the classification of triples of lattice polytopes with a given mixed volume $m$ in dimension 3. It is known that the classification can be reduced to the enumeration of so-called irreducible triples, the number of which is finite for fixed $m$. Following this algorithm, we enumerate all irreducible triples of normalized mixed volume up to 4 that are inclusion-maximal. This produces a classification of generic trivariate sparse polynomial systems with up to 4 solutions in the complex torus, up to monomial changes of variables. By a recent result of Esterov, this lea…
In 1980, V. I. Arnold studied the classification problem for convex lattice polygons of given area. Since then, this problem and its analogues have been studied by many authors, including $\mathrm{B\acute{a}r\acute{a}ny}$, Lagarias, Pach, Santos, Ziegler and Zong. Recently, Zong proposed two computer programs to prove Hadwiger's covering conjecture and Borsuk's partition problem, respectively, based on enumeration of the convex lattice polytopes contained in certain balls. For this purpose, similar to $\mathrm{B\acute{a}r\acute{a}ny}$ and Pach's work on volume and Liu and Zong's work on cardin…
Many key algorithms in 3-manifold topology involve the enumeration of normal surfaces, which is based upon the double description method for finding the vertices of a convex polytope. Typically we are only interested in a small subset of these vertices, thus opening the way for substantial optimization. Here we give an account of the vertex enumeration problem as it applies to normal surfaces, and present new optimizations that yield strong improvements in both running time and memory consumption. The resulting algorithms are tested using the freely available software package Regina.
A graph whose nodes have degree 1 or 3 is called a $\{1,3\}$-graph. Liu and Osserman associated a polytope to each $\{1,3\}$-graph and studied the Ehrhart quasi-polynomials of these polytopes. They showed that the vertices of these polytopes have coordinates in the set $\{0,\frac14,\frac12,1\}$, which implies that the period of their Ehrhart quasi-polynomials is either 1, 2, or 4. We show that the period of the Ehrhart quasi-polynomial of these polytopes is at most 2 if the graph is a tree or a cubic graph, and it is equal to 4 otherwise. In the process of proving this theorem, several interes…
We present a first example of a flag vector of a polyhedral sphere that is not the flag vector of any polytope. Namely, there is a unique 3-sphere with the parameters $(f_0,f_1,f_2,f_3;f_{02})=(12,40,40,12;120)$, but this sphere is not realizable by a convex 4-polytope. The 3-sphere, which is 2-simple and 2-simplicial, was found by Werner (2009); we present results of a computer enumeration which imply that the sphere with these parameters is unique. We prove that it is non-polytopal in two ways: First, we show that it has no oriented matroid, and thus it is not realizable; this proof was foun…
We introduce the short toric polynomial associated to a graded Eulerian poset. This polynomial contains the same information as the two toric polynomials introduced by Stanley, but allows different algebraic manipulations. The intertwined recurrence defining Stanley's toric polynomials may be replaced by a single recurrence, in which the degree of the discarded terms is independent of the rank. A short toric variant of the formula by Bayer and Ehrenborg, expressing the toric $h$-vector in terms of the $cd$-index, may be stated in a rank-independent form, and it may be shown using weighted latt…
The Monotone Upper Bound Problem asks for the maximal number M(d,n) of vertices on a strictly-increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper bound M(d,n)<=M_{ubt}(d,n) provided by McMullen's (1970) Upper Bound Theorem is tight, where M_{ubt}(d,n) is the number of vertices of a dual-to-cyclic d-polytope with n facets. It was recently shown that the upper bound M(d,n)<=M_{ubt}(d,n) holds with 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 genera…
Enumeration of max-pooling responses with generalized permutohedra
Washington University in St. Louis（圣路易斯华盛顿大学）；University of California, Riverside（加州大学河滨分校）；Harvey Mudd College（哈维穆德学院）；University of California, Los Angeles（加州大学洛杉矶分校）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）；Université du Québec à Montréal（魁北克大学蒙特利尔分校）
A parallelotope $P$ is a polytope that admits a facet-to-facet tiling of space by translation copies of $P$ along a lattice. The Voronoi cell $P_V(L)$ of a lattice $L$ is an example of a parallelotope. A parallelotope can be uniquely decomposed as the Minkowski sum of a zone closed parallelotope $P$ and a zonotope $Z(U)$, where $U$ is the set of vectors used to generate the zonotope. In this paper we consider the related question: When is the Minkowski sum of a general parallelotope and a zonotope $P+Z(U)$ a parallelotope? We give two necessary conditions and show that the vectors $U$ have to …
The enumeration of normal surfaces is a crucial but very slow operation in algorithmic 3-manifold topology. At the heart of this operation is a polytope vertex enumeration in a high-dimensional space (standard coordinates). Tollefson's Q-theory speeds up this operation by using a much smaller space (quadrilateral coordinates), at the cost of a reduced solution set that might not always be sufficient for our needs. In this paper we present algorithms for converting between solution sets in quadrilateral and standard coordinates. As a consequence we obtain a new algorithm for enumerating all sta…
A class of counting problems ask for the number of regions of a central hyperplane arrangement. By duality, this is the same as counting the vertices of a zonotope. We give several efficient algorithms, based on a linear optimization oracle, that solve this and related enumeration problems. More precisely, our algorithms compute the vertices of a zonotope from the set of its generators and inversely, recover the generators of a zonotope from its vertices. A variation of the latter algorithm also allows to decide whether a polytope, given as its vertex set, is a zonotope and when it is not, to …
