David Bremner

Primal—Dual Methods for Vertex and Facet Enumeration

Abstract. Every convex polytope can be represented as the intersection of a finite set of halfspaces and as the convex hull of its vertices. Transforming from the halfspace (resp. vertex) to the vertex (resp. halfspace) representation is called vertex enumeration (resp. facet enumeration ). An open question is whether there is an algorithm for these two problems (equivalent by geometric duality) that is polynomial in the input size and the output size. In this paper we extend the known polynomially solvable classes of polytopes by looking at the dual problems. The dual problem of a vertex (resp. facet) enumeration problem is the facet (resp. vertex) enumeration problem for the same polytope where the input and output are simply interchanged. For a particular class of polytopes and a fixed algorithm, one transformation may be much easier than its dual. In this paper we propose a new class of algorithms that take advantage of this phenomenon. Loosely speaking, primal—dual algorithms use a solution to the easy direction as an oracle to help solve the seemingly hard direction.

Necklaces, convolutions, and X+Y

We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the ℓp norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p=1, p even, and p=∞. For p even, we reduce the problem to standard convolution, while for p=∞ and p=1, we reduce the problem to (min,+) convolution and

Output-Sensitive Algorithms for Computing Nearest-Neighbour Decision Boundaries

(\operatorname {median},+)

Matroid Enumeration for Incidence Geometry

convolution. Then we solve the latter two convolution problems in subquadratic time, which are interesting results in their own right. These results shed some light on the classic sorting X+Y problem, because the convolutions can be viewed as computing order statistics on the antidiagonals of the X+Y matrix. All of our algorithms run in o(n2) time, whereas the obvious algorithms for these problems run in Θ(n2) time.

How good are convex hull algorithms

Given a set R of red points and a set B of blue points, the nearest-neighbour decision rule classifies a new point q as red (respectively, blue) if the closest point to q in R ∪ B comes from R (respectively, B). This rule implicitly partitions space into a red set and a blue set that are separated by a red-blue decision boundary. In this paper we develop output-sensitive algorithms for computing this decision boundary for point sets on the line and in ℝ2. Both algorithms run in time O(n log k), where k is the number of points that contribute to the decision boundary. This running time is the best possible when parameterizing with respect to n and k.

Edge-Graph Diameter Bounds for Convex Polytopes with Few Facets

Matroids are combinatorial abstractions for point configurations and hyperplane arrangements, which are fundamental objects in discrete geometry. Matroids merely encode incidence information of geometric configurations such as collinearity or coplanarity, but they are still enough to describe many problems in discrete geometry, which are called incidence problems. We investigate two kinds of incidence problem, the points–lines–planes conjecture and the so-called Sylvester–Gallai type problems derived from the Sylvester–Gallai theorem, by developing a new algorithm for the enumeration of non-isomorphic matroids. We confirm the conjectures of Welsh–Seymour on ≤11 points in ℝ3 and that of Motzkin on ≤12 lines in ℝ2, extending previous results. With respect to matroids, this algorithm succeeds to enumerate a complete list of the isomorph-free rank 4 matroids on 10 elements. When geometric configurations corresponding to specific matroids are of interest in some incidence problems, they should be analyzed on oriented matroids. Using an encoding of oriented matroid axioms as a boolean satisfiability (SAT) problem, we also enumerate oriented matroids from the matroids of rank 3 on n≤12 elements and rank 4 on n≤9 elements. We further list several new minimal non-orientable matroids.

Computing Constrained Minimum-Width Annuli of Point Sets

A convex polytope P can be speci ed in two ways as the convex hull of the vertex set V of P or as the intersection of the set H of its facet inducing halfspaces The vertex enumeration problem is to compute V from H The facet enumeration problem it to compute H from V These two problems are essentially equivalent under point hyperplane duality They are among the central computational problems in the theory of polytopes It is open whether they can be solved in time polynomial in jHj jVj In this paper we consider the main known classes of algorithms for solving these problems We argue that they all have at least one of two weaknesses inability to deal well with degen eracies or inability to control the sizes of intermediate results We then introduce families of polytopes that exercise those weaknesses Roughly speaking fat lattice or intricate polytopes cause algorithms with bad degeneracy handling to perform badly dwarfed polytopes cause al gorithms with bad intermediate size control to perform badly We also present computational experience with trying to solve these problem on these hard polytopes using various implementations of the main algorithms

Computing constrained minimum-width annuli of point sets

We show that the edge graph of a 6-dimensional polytope with 12 facets has diameter at most 6, thus verifying the d-step conjecture of Klee and Walkup in the case d=6. This implies that for all pairs (d, n) with n−d⩽6, the diameter of the edge graph of a d-polytope with n facets is bounded by 6, which proves the Hirsch conjecture for all n−d⩽6. We prove this result by establishing this bound for a more general structure, so-called matroid polytopes, by reduction to a small number of satisfiability problems.

Necklaces, convolutions, and X + Y

We study the problem of determining whether a manufactured disc of certain radius r is within tolerance. More precisely, we present algorithms that, given a set of n probe points on the surface of the manufactured object, compute the thinnest annulus whose outer (or inner, or median) radius is r and that contains all the probe points. Our algorithms run in O(n log n) time.

Determining the castability of simple polyhedra

We study the problem of determining whether a manufactured disk of certain radius r is within tolerance. More precisely, we present algorithms that, given a set of n probe points on the surface of the manufactured object, compute the thinnest annulus whose outer (or inner, or median) radius is r and that contains all the probe points. Our algorithms run in O(nlogn) time.