Lattice Polyhedra Research Articles

Hahn polynomials for hypergeometric distribution

Orthogonal polynomials for the multivariate hypergeometric distribution are defined on lattices in polyhedral domains in Rd. Their structures are studied through a detailed analysis of classical Hahn polynomials with negative integer parameters. Factorization of the Hahn polynomials is explored and used to explain the relationship between the index set of orthogonal polynomials and the lattice set in the polyhedral domain. In the multivariate case, these constructions lead to nontrivial families of hypergeometric polynomials vanishing on lattice polyhedra. The generating functions and bispectral properties of the orthogonal polynomials are also discussed.

Discrete volumes of lattice polyhedra via vector analysis

Pick’s theorem relates the number of lattice points to the area for a lattice polygon. Diaz and Robins gave a proof of Pick’s theorem by using the Weierstrass $$\wp $$ -function and complex analysis. As an analogue to lattice convex polyhedra, Reeve’s theorem is known as a solid version of Pick’s theorem. In this paper, we study counting lattice points on a lattice polyhedron by using vector analysis, and we extend Reeve’s theorem to nonconvex polyhedral complexes.

On generalizations of network design problems with degree bounds

Iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems. In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling more complex degree constraints in the minimum spanning tree problem (namely laminar crossing spanning tree), and (2) by incorporating ‘degree bounds’ in other combinatorial optimization problems such as matroid intersection and lattice polyhedra. We give new or improved approximation algorithms, hardness results, and integrality gaps for these problems. Our main result is a (1, b + O(log n))-approximation algorithm for the minimum crossing spanning tree (MCST) problem with laminar degree constraints. The laminar MCST problem is a natural generalization of the well-studied bounded-degree MST, and is a special case of general crossing spanning tree. We give an additive Ω(log c m) hardness of approximation for general MCST, even in the absence of costs (c > 0 is a fixed constant, and m is the number of degree constraints). This also leads to a multiplicative Ω(log c m) hardness of approximation for the robust k-median problem (Anthony et al. in Math Oper Res 35:79–101, 2010), improving over the previously known factor 2 hardness. We then consider the crossing contra-polymatroid intersection problem and obtain a (2, 2b + Δ − 1)-approximation algorithm, where Δ is the maximum element frequency. This models for example the degree-bounded spanning-set intersection in two matroids. Finally, we introduce the crossing latticep olyhedron problem, and obtain a (1, b + 2Δ − 1) approximation algorithm under certain condition. This result provides a unified framework and common generalization of various problems studied previously, such as degree bounded matroids.

Semi Regular Lattice Polyhedra

Semi Regular Lattice Polyhedra

Two-phase greedy algorithms for some classes of combinatorial linear programs

We present greedy algorithms for some classes of combinatorial packing and cover problems within the general formal framework of Hoffman and Schwartz' lattice polyhedra. Our algorithms compute in a first phase Monge solutions for the associated dual cover and packing problems and then proceed to construct greedy solutions for the primal problems in a second phase. We show optimality of the algorithms under certain sub- and supermodular assumptions and monotone constraints. For supermodular lattice polyhedra with submodular constraints, our algorithms offer the farthest reaching generalization of Edmonds' polymatroid greedy algorithm currently known.

Distributive Lattice Polyhedra

Abstract Poly-antimatroids are generalization of the notion of antimatroid to multisets. When the underlying set consists of only two elements, such two-dimensional poly-antimatroids correspond to point sets in the integer lattice Z d . In this research we concentrate on geometrical properties of two-dimensional poly-antimatroids and prove that these sets form distributive lattice polyhedra. Our findings imply that two-dimensional poly-antimatroids have convex dimension 2. Further we investigate geometrical properties of three-dimensional distributive lattice polyhedra.

On odd points and the volume of lattice polyhedra

Denote by ℤ n 3 ,n≥ 2, the lattice consisting of all pointsx in ℝ3 such thatnx belongs to the fundamental lattice ℤ3 of points with integer coordinates. Letl n be the subset of ℤ n 3 consisting of all points whose coordinates are odd multiples of 1/n. The purpose of this paper is to give several new Pick-type formulae for the volume of three-dimensional lattice polyhedra, that is, polyhedra with vertices in ℤ3. Our formulae are in terms of numbers of only thel n-points belonging to a lattice polyhedronP in contrast to already known formulae which employ numbers of all the ℤ n 3 -points inP. On our way to establishing the formulae we show that the number of points froml n belonging to a three-dimensional lattice polyhedronP has some polynomiality properties similar to those of the well-known Ehrhart polynomial expressing the number of points of ℤ n 3 inP. The paper contains also some comments on a problem of finding a volume formula which would employ only the setsl n and which would be applicable to lattice polyhedra in arbitrary dimensions.

An ‘odd’ formula for the volume of three-dimensional lattice polyhedra

Let $$\mathcal{T}$$ be the tiling of R 3 with unit cubes whose vertices belong to the fundamental lattice L 1 of points with integer coordinates. Denote by L n the lattice consisting of all points x in R 3 such that nx belongs to L 1. When the vertices of a polyhedron P in R 3 are restricted to lie in L 1 then there is a formula which relates the volume of P to the numbers of all points of two lattices L 1 and L n lying in the interior and on the boundary of P. In the simplest case of the lattices L 1 and L 2 there are 27 points in each cube from $$\mathcal{T}$$ whose relationships to the polyhedron P must be examined. In this note we present a new formula for the volume of lattice polyhedra in R 3 which involves only nine points in each cube of $$\mathcal{T}$$ : one from L 2 and eight belonging to L 4. Another virtue of our formula is that it does not employ any additional parameters, such as the Euler characteristic.

Structure-sensitivity study of partial propene oxidation over AV2P2O10 vanadium phosphate compounds

The activity and selectivity of five AV2P2O10 compounds have been examined with respect to the mild oxidation of propene at different temperatures. These materials with orthorhombic symmetry (ACa, Cd) or monoclinic symmetry (ABa, Pb, Cd) have been prepared as well crystallized samples in an evacuated silica tube. Their crystalline structure, followed by X-ray diffraction (XRD), was unchanged after the catalytic reaction and their structural properties made them ideal model catalysts to study structure-sensitivity effects. Orthorhombic compounds were more active than monoclinic ones. Catalytic results were discussed, taking into account the nature of the inserted cations, the distortion of the framework and the organization of the lattice polyhedra.

Lattice matrices, intersection of ring families and dicuts

Lattice matrices are 0/1-matrices used in the description of certain lattice polyhedra and related to dicuts in a graph. The incidence matrix A I of a so-called intersection of two ring families and the incidence matrix A D of all dicuts of a graph are examples of such matrices. After showing that any lattice matrix A can be obtained from some matrix A I by deletion or some A D by contraction, we first describe the convex hull of the rows of A I, CONV( A I), as the solution set of a system x⩾0, Bx⩽1, Rx⩽0, which is tdi. We then derive the main result, the description of CONV( A) by another tdi system. As applications, the polyhedral description of all dicuts in a graph, CONV( A D), and that of all convex sets of bounded length in a poset are established.

Shrinking Lattice Polyhedra

This paper treats the following geometric problem: Given vertices $x_1 , \cdots ,x_n $ of a polyhedron in the integer lattice in k dimensions, can that polyhedron be “shrunk” to a similar one (i.e., one whose sides remain in the same ratio as the original polyhedron) while still remaining on the integer lattice? A necessary condition is given for this to be possible, which depends on the parity of k; for $k\leqq 4$ it is shown that the condition is sufficient, and algorithms are also given to do the shrinking. In two dimensions, the algorithm only involves computing greatest common divisors over the Gaussian integers and is polynomial time. For $k = 3$ and 4 the algorithms involve computing g.c.d.s in the algebra of Hurwitz quaternions. This gives a polynomial time algorithm for $k = 4$, but because the algorithm in three dimensions relies on determining the square factors of an integer, it is at present exponential. The proofs are remarkably simple and are quite computational in nature.

Electron-phonon interactions in bct white tin

The electron-phonon interactions are evaluated exactly over the actual shape of the atomic polyhedron as well as the lattice polyhedron of diatomic white tin by making use of simple coordinate axes transformations and crystal symmetry. It is shown that the expressions for the interference factor,S(q, t) of the atomic polyhedron are complex while those for the lattice polyhedron are real and the reciprocal lattice vectors derived from the former do not correspond to those derived either from the latter or from x-ray structure factors. By comparing these expressions with each other as well as with those obtained by approximating these polyhedra by an ellipsoid of equivalent volume, apparent differences between the interference factors of atomic and lattice polyhedra, consequent ambiguity regarding the shape and size of the first Brillouin zone of white tin, validity of the Wigner-Seitz approximation for a diatomic lattice and the manner in which the electron-phonon interactions contribute to acoustical and optical modes of vibration are discussed.

Path-closed sets

Given a digraphG = (V, E), call a node setT ⊆V path-closed ifv, v′ eT andw eV is on a path fromv tov′ impliesw eT. IfG is the comparability graph of a posetP, the path-closed sets ofG are the convex sets ofP. We characterize the convex hull of (the incidence vectors of) all path-closed sets ofG and its antiblocking polyhedron inR v , using lattice polyhedra, and give a minmax theorem on partitioning a given subset ofV into path-closed sets. We then derive good algorithms for the linear programs associated to the convex hull, solving the problem of finding a path-closed set of maximum weight sum, and prove another min-max result closely resembling Dilworth’s theorem.

A Further Note on the Volume of Lattice Polyhedra

A Further Note on the Volume of Lattice Polyhedra