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.     

Hamiltonicity in (0–1)-polyhedra

The graph G(P) of a polyhedron P has a node corresponding to each vertex of P and two nodes are adjacent in G(P) if and only if the corresponding vertices of P are adjacent on P. We show that if P ? $\text{R}$ⁿ is a polyhedron, all of whose vertices have (0–1)-valued coordinates, then (i) if G(P) is bipartite, the G(P) is a hypercube; (ii) if G(P) is nonbipartite, then G(P) is hamilton connected. It is shown that if P ? $\text{R}$ⁿ has (0–1)-valued vertices and is of dimension d (≤n) then there exists a polyhedron P′ ? $\text{R}$^d having (0–1)-valued vertices such that $\text{G(P) ? G(P′)}$. Some combinatorial consequences of these results are also discussed.     

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 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.     

Monochromatic square-cycle and square-path partitions

The square of a path P is the graph obtained from P by joining every pair of vertices with distance 2 in P. We show that in every 2-colored complete graph on sufficiently many vertices the vertex set can be partitioned by at most 65 vertex disjoint monochromatic square-paths.     

On polyhedra induced by point sets in space

Given a set S of n?3 points in the plane (not all on a line) it is well known that it is always possible to polygonize S, i.e., construct a simple polygon P such that the vertices of P are precisely the given points in S. For example, the shortest circuit through S is a simple polygon. In 1994, Grünbaum showed that an analogous theorem holds in R³. More precisely, if S is a set of n?4 points in R³ (not all of which are coplanar) then it is always possible to polyhedronize S, i.e., construct a simple (sphere-like) polyhedron P such that the vertices of P are precisely the given points in S. Grünbaum's constructive proof may yield Schönhardt polyhedra that cannot be triangulated. In this paper several alternative algorithms are proposed for constructing such polyhedra induced by a set of points, which may always be triangulated, and which enjoy several other useful properties as well. Such properties include polyhedra that are star-shaped, have Hamiltonian skeletons, and admit efficient point-location queries. We show that polyhedronizations with a variety of such useful properties can be computed efficiently in O(nlogn) time. Furthermore, we show that a tetrahedralized, xy-monotonic, polyhedronization of S may be computed in time O(n^1+ε), for any ε>0.     

Representation complexity of adaptive 3D distance fields

In this paper we study the representation complexity of a kind of data structure that stores the information necessary to compute the distance from a point to a geometric body. These data structures called adaptive splitting based on cubature distance fields (ASBCDF), are binary search trees generated by the adaptive splitting based on cubature (ASBC) algorithm that adaptively subdivides the space surrounding the body into tetrahedra. Their representation complexity is measured by the number of nodes in the tree (two times the number of tetrahedra in the resulting tessellation). In the case of convex polyhedra we prove that this quantity remains bounded as the number of vertices of the polyhedra increases to infinity. Experimental results show that the number of tetrahedra in the tessellations is almost independent of the combinatorial complexity of the polyhedra. This means that the average compute time of the distance to arbitrary convex polyhedra is almost constant. Therefore, ASBCDFs are especially suitable for real time applications involving rapidly changing environments modelized with complex polyhedra.     

Edge-Directions of Standard Polyhedra with Applications to Network Flows

Recent results show that edge-directions of polyhedra play an important role in (combinatorial) optimization; in particular, a d-dimensional polyhedron with |D| distinct edge-directions has at most O(|D|^d-1) vertices. Here, we obtain a characterization of the directions of edges that are adjacent to a given vertex of a standard polyhedron of the form P = {x : Ax = b, l⩽ x⩽ u, tightening a standard necessary condition which asserts that such directions must be minimal support solutions of the homogenous equation Ax = 0 which are feasible at the given vertex. We specialize the characterization for polyhedra that correspond to network flows, obtaining a graph characterization of circuits which correspond to edge-directions. Applications to partitioning polyhedra are discussed. Research of these authors was supported by a grant from ISF – the Israel Science Foundation, by a VPR grant at the Technion, and by the Fund for the Promotion of Research at the Technion.     

Remarkable polyhedra related to set functions,games and capacities

Set functions are widely used in many domains of operations research (cooperative game theory, decision under risk and uncertainty, combinatorial optimization) under different names (TU-game, capacity, nonadditive measure, pseudo-Boolean function, etc.). Remarkable families of set functions form polyhedra, e.g., the polytope of capacities, the polytope of p-additive capacities, the cone of supermodular games, etc. Also, the core of a set function, defined as the set of additive set functions dominating that set function, is a polyhedron which is of fundamental importance in game theory, decision-making and combinatorial optimization. This survey paper gives an overview of these notions and studies all these polyhedra.     

Polybasic polyhedra: structure of polyhedra with edge vectors of support size at most 2

It has widely been recognized that submodular set functions and base polyhedra associated with them play fundamental and important roles in combinatorial optimization problems. In the present paper, we introduce a generalized concept of base polyhedron. We consider a class of pointed convex polyhedra in R^V whose edge vectors have supports of size at most 2. We call such a convex polyhedron a polybasic polyhedron. The class of polybasic polyhedra includes ordinary base polyhedra, submodular/supermodular polyhedra, generalized polymatroids, bisubmodular polyhedra, polybasic zonotopes, boundary polyhedra of flows in generalized networks, etc. We show that for a pointed polyhedron PR^V, the following three statements are equivalent:

(1) P is a polybasic polyhedron.

(2) Each face of P with a normal vector of the full support V is obtained from a base polyhedron by a reflection and scalings along axes.

(3) The support function of P is a submodular function on each orthant of R^V.

This reveals the geometric structure of polybasic polyhedra and its relation to submodularity.     

Closed-Form Expressions for Uniform Polyhedra and Their Duals

A uniform polyhedron has equivalent vertices and regular polygonal faces. An established set of 77 Wythoff symbols effectively describes the dynamic kaleidoscopic constructions of uniform polyhedra. The main combinatorial and metrical quantities of uniform polyhedra and their duals are presented as closed-form expressions derived from the Wythoff symbols. Received September 20, 2000, and in revised form September 20, 2001. Online publication February 19, 2002.     

Quotient polytopes of cyclic polytopes part I: Structure and characterization

We investigate the quotient polytopesC/F, whereC is a cyclic polytope andF is a face ofC. We describe the combinatorial structure of such quotients, and show that under suitable restrictions the pair (C, F) is determined by the combinatorial type ofC/F. We describe alternative constructions of these quotients by “splitting vertices” of lower-dimensional cyclic polytopes. Using Gale diagrams, we show that every simpliciald-polytope withd+3 vertices is isomorphic to a quotient of a cyclic polytope.     

Morphing polyhedra with parallel faces: Counterexamples

Two simple polyhedra P and Q (not necessarily convex) are parallel if they share the same edge graph G and each face of P has the same outward-facing unit normal as the corresponding face in Q. Parallel polyhedra P and Q admit a parallel morph if the vertices can be moved in a continuous manner taking us from P to Q such that at all times the intermediate polyhedron determined by the vertex configuration and graph G is both simple and parallel with P (and Q). In this note, we show that even for very restrictive classes of orthogonal polyhedra, there exist parallel polyhedra that do not admit a parallel morph.     

Greedoid polyhedra

Greedoids can be viewed as relaxations of matroids. As a subclass, APS-greedoids cover many combinatorial problems. This paper deals with APS-greedoid polyhedra. These polyhedra are similar in properties to matroid polyhedra. That is, the vertices of an APS-greedoid polyhedron are precisely the incidence vectors of the members of an APS-greedoid.Graduate School of USTC     

Decompositions and boundary coverings of non-convex fat polyhedra

We show that any locally-fat (or (α,β)-covered) polyhedron with convex fat faces can be decomposed into O(n) tetrahedra, where n is the number of vertices of the polyhedron. We also show that the restriction that the faces are fat is necessary: there are locally-fat polyhedra with non-fat faces that require Ω(n²) pieces in any convex decomposition. Furthermore, we show that if we want the tetrahedra in the decomposition to be fat themselves, then their number cannot be bounded as a function of n in the worst case. Finally, we obtain several results on the problem where we want to only cover the boundary of the polyhedron, and not its entire interior.     

An algebraic and combinatorial approach to the analysis of line drawings of polyhedra

Mathematical structures of line drawings of polyhedral scenes are studied from the viewpoint of scene analysis. First, algebraic structures of line drawings are elucidated, and a necessary and sufficient condition is obtained for a line drawing to represent a polyhedron. Next, combinatorial structures are investigated and the class of pictures that represent nontrivial three-dimensional configurations when vertices are drawn in general position is characterized by incidence structures of polyhedra. The results are furthermore applied to correction of vertex-position errors, discrimination between correct and incorrect line drawings, recognition of unique solvability of some figure-construction problems, classification of line drawings, and other related problems.     

Complete subgraphs are elusive

We would like to decide whether a graph with a given vertex set has a certain property. We do this by probing the pairs of vertices one by one, i.e., asking whether a given pair of vertices is an edge or not. At each stage we make full use of the information we have up to that point. If there is an algorithm (a sequence of probes depending on the previous information) that allows us to come to a decision before checking every pair, the property is said to be incomplete, otherwise it is called complete or elusive. We show that the property of containing a complete subgraph with a given number of vertices is elusive. The proof also implies that the property of being r-chromatic (r fixed) is elusive.     

Generating All Vertices of a Polyhedron Is Hard

We show that generating all negative cycles of a weighted graph is a hard enumeration problem, in both the directed and undirected cases. More precisely, given a family of negative (directed) cycles, it is an NP-complete problem to decide whether this family can be extended or there are no other negative (directed) cycles in the graph, implying that (directed) negative cycles cannot be generated in polynomial output time, unless P=NP. As a corollary, we solve in the negative two well-known generating problems from linear programming: (i) Given an infeasible system of linear inequalities, generating all minimal infeasible subsystems is hard. Yet, for generating maximal feasible subsystems the complexity remains open. (ii) Given a feasible system of linear inequalities, generating all vertices of the corresponding polyhedron is hard. Yet, in the case of bounded polyhedra the complexity remains open. Equiva lently, the complexity of generating vertices and extreme rays of polyhedra remains open.     

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.