Family Of Polytopes Research Articles

A new formula for the determinant and bounds on its tensor and Waring ranks

Abstract We present a new explicit formula for the determinant that contains superexponentially fewer terms than the usual Leibniz formula. As an immediate corollary of our formula, we show that the tensor rank of the $n \times n$ determinant tensor is no larger than the $n$ -th Bell number, which is much smaller than the previously best-known upper bounds when $n \geq 4$ . Over fields of non-zero characteristic we obtain even tighter upper bounds, and we also slightly improve the known lower bounds. In particular, we show that the $4 \times 4$ determinant over ${\mathbb{F}}_2$ has tensor rank exactly equal to $12$ . Our results also improve upon the best-known upper bound for the Waring rank of the determinant when $n \geq 17$ , and lead to a new family of axis-aligned polytopes that tile ${\mathbb{R}}^n$ .

Facet separation for disjunctive constraints with network flow representation

We present a novel algorithm for separating facet-inducing inequalities for the convex-hull of the union of polytopes representing a disjunctive constraint of special structure. It is required that the union of polytopes admit a certain network flow representation. The algorithm is based on a new, graph theoretic characterization of the facets of the convex-hull. Moreover, we characterize the family of polytopes that are representable by the networks under consideration. We demonstrate the effectiveness of our approach by computational results on a set of benchmark problems.

On hendecagonal circular ladder and its metric dimension

Let Γ = ( V , E ) be a connected graph of order n. Two vertices p and q in V are said to resolve by a vertex x ∈ V if d ( p , x ) ≠ d ( q , x ) . An ordered subset F = { k 1 , k 2 , k 3 , … , k l } of vertices in Γ is said to be resolving set if for every pair p, q of distinct vertices in V, we have ζ ( p | F ) ≠ ζ ( q | F ) , where ζ ( a | F ) = ( d ( a , k 1 ) , d ( a , k 2 ) , d ( a , k 3 ) , … , d ( a , k l ) ) is the l-code/metric coordinate representation of the vertex a with respect to the set F. The resolving set for Γ with minimum cardinality is known as metric basis for the graph Γ and the cardinality of metric basis is called as metric dimension of Γ. In this work, we demonstrate that for two families of convex polytopes which are closely linked, the metric dimension is constant.

Modified Detour Index of Hamiltonian Connected (Laceable) Graphs

Objectives: To explore the bounds for the modified detour index of certain Hamiltonian connected and laceable graphs. Methods: The Wiener index , detour index and the modified detour index are used. Findings: Here we introduce the modified detour index and its least upper bounds for Hamiltonian connected and laceable graphs, by formulating the constraints. Novelty: Based on the modified detour index, the bounds for some special graphs such as: Hamiltonian connected graphs of two families of convex polytopes ( and ) and Hamiltonian laceable graphs of spider graph ( ) and image graph of prism graph ( ) are encountered here. Keywords: Hamiltonian graph, Hamiltonian connected, Hamiltonian laceable, Wiener index, detour index

Relative poset polytopes and semitoric degenerations

The two best studied toric degenerations of the flag variety are those given by the Gelfand–Tsetlin and FFLV polytopes. Each of them degenerates further into a particular monomial variety which raises the problem of describing the degenerations intermediate between the toric and the monomial ones. Using a theorem of Zhu one may show that every such degeneration is semitoric with irreducible components given by a regular subdivision of the corresponding polytope. This leads one to study the parts that appear in such subdivisions as well as the associated toric varieties. It turns out that these parts lie in a certain new family of poset polytopes which we term relative poset polytopes: each is given by a poset and a weakening of its order relation. In this paper we give an in depth study of (both common and marked) relative poset polytopes and their toric varieties in the generality of an arbitrary poset. We then apply these results to degenerations of flag varieties. We also show that our family of polytopes generalizes the family studied in a series of papers by Fang, Fourier, Litza and Pegel while sharing their key combinatorial properties such as pairwise Ehrhart-equivalence and Minkowski-additivity.

Locating-dominating number of certain infinite families of convex polytopes with applications

A convex hull of finitely many points in the Euclidean space Rd is known as a convex polytope. Graphically, they are planar graphs i.e. embeddable on R2. Minimum dominating sets possess diverse applications in computer science and engineering. Locating-dominating sets are a natural extension of dominating sets. Studying minimizing locating-dominating sets of convex polytopes reveal interesting distance-dominating related topological properties of these geometrical planar graphs. In this paper, exact value of the locating-dominating number is shown for one infinite family of convex polytopes. Moreover, tight upper bounds on γl−d are shown for two more infinite families. Tightness in the upper bounds is shown by employing an updated integer linear programming (ILP) model for the locating-dominating number γl−d of a fixed graph. Results are explained with help of some examples. The second part of the paper solves an open problem in Khan (2023) [28] which asks to find a domination-related parameter which delivers a correlation coefficient of ρ>0.9967 with the total π-electronic energy of lower benzenoid hydrocarbons. We show that the locating-dominating number γl−d delivers such a strong prediction potential. The paper is concluded with putting forward some open problems in this area.

On polytopes with linear rank with respect to generalizations of the split closure

In this paper we study the rank of polytopes contained in the 0-1 cube with respect to t-branch split cuts and t-dimensional lattice cuts for a fixed positive integer t. These inequalities are the same as split cuts when t=1 and generalize split cuts when t>1. For polytopes contained in the n-dimensional 0-1 cube, the work of Balas implies that the split rank can be at most n, and this bound is tight as Cornuéjols and Li gave an example with split rank n. All known examples with high split rank – i.e., at least cn for some positive constant c<1 – are defined by exponentially many (as a function of n) linear inequalities. For any fixed integer t>0, we give a family of polytopes contained in [0,1]n for sufficiently large n such that each polytope has empty integer hull, is defined by O(n) inequalities, and has rank Ω(n) with respect to t-dimensional lattice cuts. Therefore the split rank of these polytopes is Ω(n). It was shown earlier that there exist generalized branch-and-bound proofs, with logarithmic depth, of the nonexistence of integer points in these polytopes. Therefore, our lower bound results on split rank show an exponential separation between the depth of branch-and-bound proofs and split rank.

Cardinality bounds on subsets in the partition resolving set for complex convex polytope-like graph

&lt;abstract&gt;&lt;p&gt;Let $ G = (V, E) $ be a simple, connected graph with vertex set $ V(G) $ and $ E(G) $ edge set of $ G $. For two vertices $ a $ and $ b $ in a graph $ G $, the distance $ d(a, b) $ from $ a $ to $ b $ is the length of shortest path $ a-b $ path in $ G $. A $ k $-ordered partition of vertices of $ G $ is represented as $ {R}{p} = \{{R}{p_1}, {R}{p_2}, \dots, {R}{p_k}\} $ and the representation $ r(a|{R}{p}) $ of a vertex $ a $ with respect to $ {R}{p} $ is the vector $ (d(a|{R}{p_1}), d(a|{R}{p_2}), \dots, d(a|{R}{p_k})) $. The partition is called a resolving partition of $ G $ if $ r(a|{R}{p}) \ne r(b|{R}{p}) $ for all distinct $ a, b\in V(G) $. The partition dimension of a graph, denoted by $ pd(G) $, is the cardinality of a minimum resolving partition of $ G $. Computing precise and constant values for the partition dimension poses a interesting problem; therefore, it is possible to compute an upper bound for the partition dimension within a general family of graphs. In this paper, we studied partition dimension of the some families of convex polytopes, specifically $ \mathbb{T}_n $, $ \mathbb{U}_n $, $ \mathbb{V}_n $, and $ \mathbb{A}_n $, and proved that these graphs have constant partition dimension.&lt;/p&gt;&lt;/abstract&gt;

On metric dimension of hendecagonal circular ladder $H_{n}$

Let $\zeta=(V,E)$ be a $n$th order connected graph. If the distance vectors to the vertices in an ordered subset $G$ of vertices can uniquely identify each vertex of the graph $\zeta$, then the set $G$ is known as resolving set for the graph $\zeta$. The resolving set $G$ with smallest cardinality serves as the metric dimension of graph $\zeta$ and this resolving set serves as the metric basis for $\zeta$. In this article, two families of convex polytopes that are closely linked are demonstrated and it is found that the metric dimension is three for both the families.

Angle Sums of Random Polytopes

For two families of random polytopes, we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any given dimension of the polytope under consideration. As particular cases, we compute the expected sums of internal and external angles at all faces of any fixed dimension. The first family is the Gaussian polytopes defined as convex hulls of i.i.d. samples from a nondegenerate Gaussian distribution in Rd. The second family is convex hulls of random walks with exchangeable increments satisfying certain mild general position assumption. The expected sums are expressed in terms of the angles of the regular simplices and the Stirling numbers, respectively. There are nontrivial analogies between these two settings. Further, we compute the angle sums for Gaussian projections of arbitrary polyhedral sets, of which the Gaussian polytopes are a particular case. Also, we show that the expected Grassmann angle sums of a random polytope with a rotationally invariant law are invariant under affine transformations. Of independent interest may be also results on the faces of linear images of polyhedral sets. These results are well known, but it seems that no detailed proofs can be found in the existing literature.

A triangulation for pointed order polytopes

In this paper we propose a way to triangulate a pointed order polytope. Pointed order polytopes are a generalization of order polytopes that include some important groups of polytopes appearing when bipolar scales arise in Decision Making or Game Theory, as the set of bi-capacities or the set of normalized bi-games, even for cases with restricted cooperation. Triangulating polytopes is an important and difficult problem that allows an elegant way to generate uniform random points in the polytope. For order polytopes, there exists a nice result that allows a way to triangulate this family of polytopes based on generating linear extensions. In this paper we prove a similar result for pointed order polytopes. The results in this paper allow to derive a procedure to generate random points inside a pointed order polytope that depends only on the structure of the subjacent poset, a problem that usually is simpler to tackle. In particular, this could be applied to generate bi-capacities or bi-capacities belonging to some subfamilies (e.g. k-symmetric, k-interactive, ...) in a random way.

On vertex resolvability of a circular ladder of nonagons

Let $H=H(V,E)$ be a non-trivial simple connected graph with edge and vertex set $E(H)$ and $V(H)$, respectively. A subset $\mathbb{D}\subset V(H)$ with distinct vertices is said to be a vertex resolving set in $H$ if for each pair of distinct vertices $p$ and $q$ in $H$ we have $d(p,u)\neq d(q,u)$ for some vertex $u\in H$. A resolving set $H$ with minimum possible vertices is said to be a metric basis for $H$. The cardinality of metric basis is called the metric dimension of $H$, denoted by $\dim_{v}(H)$. In this paper, we prove that the metric dimension is constant and equal to $3$ for certain closely related families of convex polytopes.

The equivariant Ehrhart theory of polytopes with order-two symmetries

We study the equivariant Ehrhart theory of families of polytopes that are invariant under a non-trivial action of the group with order two. We study families of polytopes whose equivariant H ∗ H^\ast -polynomial both succeed and fail to be effective, in particular, the symmetric edge polytopes of cycle graphs and the rational cross-polytope. The latter provides a counterexample to the effectiveness conjecture if the requirement that the vertices of the polytope have integral coordinates is loosened to allow rational coordinates. Moreover, we exhibit such a counterexample whose Ehrhart function has period one and coincides with the Ehrhart function of a lattice polytope.

Infinite Families of Hypertopes from Centrally Symmetric Polytopes

We construct infinite families of abstract regular polytopes of Schläfli type $\{4,p_1,\ldots,p_{n-1}\}$ from extensions of centrally symmetric spherical abstract regular $n$-polytopes. In addition, by applying the halving operation, we obtain infinite families of both locally spherical and locally toroidal regular hypertopes of type $\left\{\genfrac{}{}{0pt}{}{p_1}{p_1},\ldots,p_{n-1}\right\}$.

Chain enumeration, partition lattices and polynomials with only real roots

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 barycentric subdivisions have real-rooted \(f\)-polynomials are presented. An application to the face enumeration of the second barycentric subdivision of the boundary complex of the simplex is also included.Mathematics Subject Classifications: 05A05, 05A18, 05E45, 06A07, 26C10Keywords: Chain polynomial, geometric lattice, partition lattice, real-rooted polynomial, flag \(h\)-vector, convex polytope, barycentric subdivision

On the beta distribution, the nonlinear Fourier transform and a combinatorial problem

The paper describes some probabilistic and combinatorial aspects of the nonlinear Fourier transform associated with the AKNS-ZS problems. One of the two main results shows that the volumes of a family of polytopes that appear in a power expansion of the nonlinear Fourier transform are distributed according to the beta probability distribution. We establish this result by studying an Euler-type discretisation of the nonlinear Fourier transform. This approach provides another main result. We discover a novel discrete probability distribution approximating the beta distribution with integral shape parameters. For specific choices of the shape parameters, our new discrete distribution has a natural combinatorial interpretation; the numbers of alternating ordered partitions of an integer into distinct parts are essentially distributed according to our new discrete distribution. The generating function for these numbers can be easily expressed using the nonlinear Fourier transform of the constant function.

Generalized Permutahedra and Optimal Auctions

We study a family of convex polytopes, called SIM-bodies, which were introduced by Giannakopoulos and Koutsoupias in 2018 to analyze so-called Straight-Jacket Auctions. First, we show that the SIM-bodies belong to the class of generalized permutahedra. Second, we prove an optimality result for the Straight-Jacket Auctions among certain deterministic auctions. Third, we employ computer algebra methods and mathematical software to explicitly determine optimal prices and revenues.

Partial Permutation and Alternating Sign Matrix Polytopes

We define and study a new family of polytopes which are formed as convex hulls of partial alternating sign matrices. We determine the inequality descriptions, number of facets, and face lattices of these polytopes. We also study partial permutohedra that we show arise naturally as projections of these polytopes. We enumerate facets and also characterize the face lattices of partial permutohedra in terms of chains in the Boolean lattice. Finally, we have a result and a conjecture on the volume of partial permutohedra when one parameter is fixed to be two.

Families of Polytopes with Rational Linear Precision in Higher Dimensions

In this article, we introduce a new family of lattice polytopes with rational linear precision. For this purpose, we define a new class of discrete statistical models that we call multinomial staged tree models. We prove that these models have rational maximum likelihood estimators (MLE) and give a criterion for these models to be log-linear. Our main result is then obtained by applying Garcia-Puente and Sottile’s theorem that establishes a correspondence between polytopes with rational linear precision and log-linear models with rational MLE. Throughout this article, we also study the interplay between the primitive collections of the normal fan of a polytope with rational linear precision and the shape of the Horn matrix of its corresponding statistical model. Finally, we investigate lattice polytopes arising from toric multinomial staged tree models, in terms of the combinatorics of their tree representations.

The diagonal of the operahedra

The primary goal of this article is to set up a general theory of coherent cellular approximations of the diagonal for families of polytopes by developing the method introduced by N. Masuda, A. Tonks, H. Thomas and B. Vallette. We apply this theory to the study of the operahedra, a family of polytopes ranging from the associahedra to the permutahedra, and which encodes homotopy operads. After defining Loday realizations of the operahedra, we make a coherent choice of cellular approximations of the diagonal, which leads to a compatible topological cellular operad structure on them. This gives a model for topological and algebraic homotopy operads and an explicit functorial formula for their tensor product.