On the characterization of noninteger vertices of the relaxation polyhedron in the multi-index axial assignment problem

Theorems about the characterization and exponential growth of the denominators of fractional components of noninteger vertices of the relaxation polyhedron in the multi-index axial assignment problem are proved. They made it possible to obtain new lower bounds on the number of noninteger vertices of this polyhedron and to refute the conjecture on the estimate of the ratio of the number of integer vertices to the number of all vertices of the multi-index axial transportation polyhedron determined by integer vectors.     

Types of maximum noninteger vertices of the relaxation polyhedron of the four-index axial assignment problem

We describe various types of maximum noninteger vertices. We identify types of polyhedron vertices by the number of fractional components contained in three-sections of fourindex matrices representing the polyhedron vertices.     

Vertex-faithful regular polyhedra

We study the abstract regular polyhedra with automorphism groups that act faithfully on their vertices, and show that each non-flat abstract regular polyhedron covers a “vertex-faithful” polyhedron with the same number of vertices. We then use this result and earlier work on flat polyhedra to study abstract regular polyhedra based on the size of their vertex set. In particular, we classify all regular polyhedra where the number of vertices is prime or twice a prime. We also construct the smallest regular polyhedra with a prime squared number of vertices.     

On rainbowness of semiregular polyhedra

We introduce the rainbowness of a polyhedron as the minimum number k such that any colouring of vertices of the polyhedron using at least k colours involves a face all vertices of which have different colours. We determine the rainbowness of Platonic solids, prisms, antiprisms and ten Archimedean solids. For the remaining three Archimedean solids this parameter is estimated. Dedicated to Professor Miroslav Fiedler on the occasion of his 80th birthday.     

On the difficulty of triangulating three-dimensional Nonconvex Polyhedra

A number of different polyhedraldecomposition problems have previously been studied, most notably the problem of triangulating a simple polygon. We are concerned with thepolyhedron triangulation problem: decomposing a three-dimensional polyhedron into a set of nonoverlapping tetrahedra whose vertices must be vertices of the polyhedron. It has previously been shown that some polyhedra cannot be triangulated in this fashion. We show that the problem of deciding whether a given polyhedron can be triangulated is NP-complete, and hence likely to be computationally intractable. The problem remains NP-complete when restricted to the case of star-shaped polyhedra. Various versions of the question of how many Steiner points are needed to triangulate a polyhedron also turn out to be NP-hard.This work was supported by National Science Foundation Grant CCR-8809040.     

A polyhedron of genus 4 with minimal number of vertices and maximal symmetry

We construct a symmetric polyhedron of genus 4 in R ³ with 11 vertices. This shows that for given genus g the minimal numbers of vertices of combinatorial manifolds and of polyhedra coincide in the first previously unknown case g=4 also. We show that our polyhedron has the maximal symmetry for the given genus and minimal number of vertices.     

On nonnegative solutions of random systems of linear inequalities

The set of nonnegative solutions of a system of linear equations or inequalities is a convex polyhedron. If the coefficients of the system are chosen at random, the number of vertices of this polyhedron is a random variable. Its expected value, dependent on the probability distribution of the coefficients, which are assumed to be nonnegative throughout, is investigated, and a distribution-independent upper bound for this expected value is established.     

Combinatorial structure and adjacency of vertices of polytope of b-factors

In the present paper in terms of the graph theory we describe the structure and vertices adjacency criterion of b-factors polyhedron. The special attention is paid to nonintegral vertices. Results of the present paper, in particular, generalize properties of nonintegral vertices of TSP polyhedron, give vertices adjacency criterion of a transportation polytope.     

Reconstructing orthogonal polyhedra from putative vertex sets

In this paper we study the problem of reconstructing orthogonal polyhedra from a putative vertex set, i.e., we are given a set of points and want to find an orthogonal polyhedron for which this is the set of vertices. This is well-studied in 2D; we mostly focus on 3D, and on the case where the given set of points may be rotated beforehand. We obtain fast algorithms for reconstruction in the case where the answer must be orthogonally convex.     

Bounds on the number of vertices of perturbed polyhedra

Finding the incident edges to a degenerate vertex of a polyhedron is a non-trivial problem. So pivoting methods generally involve a perturbation argument to overcome the degeneracy problem. But the perturbation entails a bursting of each degenerate vertex into a cluster of nondegenerate vertices. The aim of this paper is to give some bounds on the number of these perturbed vertices.     

Determination of the efficient set in multiobjective linear programming

This paper develops a method for finding the whole set of efficient points of a multiobjective linear problem. Two algorithms are presented; the first one describes the set of all efficient vertices and all efficient rays of the constraint polyhedron, while the second one generates the set of all efficient faces. The method has been tested on several examples for which numerical results are reported.The authors are grateful to Professor W. Stadler and an anonymous referee for their helpful comments and corrections.     

Minimal Ellipsoid Circumscribing a Polytope Defined by a System of Linear Inequalities

In this paper, we will propose algorithms for calculating a minimal ellipsoid circumscribing a polytope defined by a system of linear inequalities. If we know all vertices of the polytope and its cardinality is not very large, we can solve the problem in an efficient manner by a number of existent algorithms. However, when the polytope is defined by linear inequalities, these algorithms may not work since the cardinality of vertices may be huge. Based on a fact that vertices determining an ellipsoid are only a fraction of these vertices, we propose algorithms which iteratively calculate an ellipsoid which covers a subset of vertices. Numerical experiment shows that these algorithms perform well for polytopes of dimension up to seven.     

Master corner polyhedron: Vertices

We focus on the vertices of the master corner polyhedron (MCP), a fundamental object in the theory of integer linear programming. We introduce two combinatorial operations that transform vertices to their neighbors. This implies that each MCP can be defined by the initial vertices regarding these operations; we call them support vertices. We prove that the class of support vertices of all MCPs over a group is invariant under automorphisms of this group and describe MCP vertex bases. Among other results, we characterize its irreducible points, establish relations between a vertex and the nontrivial facets that pass through it, and prove that this polyhedron is of diameter 2.     

A Rigidity Criterion for Non-Convex Polyhedra

Let P be a (non-necessarily convex) embedded polyhedron in R³, with its vertices on the boundary of an ellipsoid. Suppose that the interior of $P$ can be decomposed into convex polytopes without adding any vertex. Then P is infinitesimally rigid. More generally, let P be a polyhedron bounding a domain which is the union of polytopes C₁, . . ., C_n with disjoint interiors, whose vertices are the vertices of P. Suppose that there exists an ellipsoid which contains no vertex of P but intersects all the edges of the C_i. Then P is infinitesimally rigid. The proof is based on some geometric properties of hyperideal hyperbolic polyhedra.     

A polyhedron with alls—t cuts as vertices,and adjacency of cuts

Consider the polyhedron represented by the dual of the LP formulation of the maximums–t flow problem. It is well known that the vertices of this polyhedron are integral, and can be viewed ass–t cuts in the given graph. In this paper we show that not alls–t cuts appear as vertices, and we give a characterization. We also characterize pairs of cuts that form edges of this polyhedron. Finally, we show a set of inequalities such that the corresponding polyhedron has as its vertices alls–t cuts.Work done at the Department of Computer Science and Engineering, Indian Institute of Technology, Delhi, India.     

The pairwise flowtime network construction problem

It is required to find an optimal order of constructing the edges of a network so as to minimize the sum of the weighted connection times of relevant pairs of vertices. Construction can be performed anytime anywhere in the network, with a fixed overall construction speed. The problem is strongly NP-hard even on stars. We present polynomial algorithms for the problem on trees with a fixed number of leaves, and on general networks with a fixed number of relevant pairs.     

Unfolding Orthogonal Polyhedra with Quadratic Refinement: The Delta-Unfolding Algorithm

We show that every orthogonal polyhedron homeomorphic to a sphere can be unfolded without overlap while using only polynomially many (orthogonal) cuts. By contrast, the best previous such result used exponentially many cuts. More precisely, given an orthogonal polyhedron with n vertices, the algorithm cuts the polyhedron only where it is met by the grid of coordinate planes passing through the vertices, together with Θ(n ²) additional coordinate planes between every two such grid planes.     

Distributed Online Frequency Assignment in Cellular Networks

A cellular network is generally modeled as a subgraph of the triangular lattice. The distributed online frequency assignment problem can be abstracted as a multicoloring problem on a weighted graph, where the weight vector associated with the vertices models the number of calls to be served at the vertices and is assumed to change over time. In this paper, we develop a framework for studying distributed online frequency assignment in cellular networks. We present the first distributed online algorithms for this problem with proven bounds on their competitive ratios. We show a series of algorithms that use at each vertex information about increasingly larger neighborhoods of the vertex, and that achieve better competitive ratios. In contrast, we show lower bounds on the competitive ratios of some natural classes of online algorithms.