Related papers: The numerical algebraic geometry of bottlenecks
Computational topology is an area that revisits topological problems from an algorithmic point of view, and develops topological tools for improved algorithms. We survey results in computational topology that are concerned with graphs drawn…
The field of numerical algebraic geometry consists of algorithms for numerically solving systems of polynomial equations. When the system is exact, such as having rational coefficients, the solution set is well-defined. However, for a…
This paper studies binary quadratic programs in which the objective is defined by a Euclidean distance matrix, subject to a general polyhedral constraint set. This class of nonconcave maximisation problems includes the capacitated,…
Studies on time and memory costs of products in geometric algebra have been limited to cases where multivectors with multiple grades have only non-zero elements. This allows to design efficient algorithms for a generic purpose; however, it…
Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…
When three-dimensional bodies contain thin features, non-trivial topology, or scan-derived surfaces, volumetric meshing can become the dominant bottleneck in simulation workflows. We replace this step with a learned geometric…
Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NP-hard optimization problems on unit disk graphs. The problems considered include…
We study the geometry of algebraic numbers in the complex plane, and their Diophantine approximation, aided by extensive computer visualization. Motivated by these images, called algebraic starscapes, we describe the geometry of the map…
Covering numbers are a powerful tool used in the development of approximation algorithms, randomized dimension reduction methods, smoothed complexity analysis, and others. In this paper we prove upper bounds on the covering number of…
We introduce two new algebraic invariants, the (co)homological distances between continuous maps, which provide computable lower bounds for the homotopic distance and strictly refine the classical cup-length estimates. We then define the…
The problem of approximating the infinite dimensional space of all continuous maps from an algebraic variety $X$ to an algebraic variety $Y$ by finite dimensional spaces of algebraic maps arises in several areas of geometry and mathematical…
We consider the distance minimization problem to a real algebraic variety $X \subseteq \RR^n$ when the metric is induced by a polyhedral norm. Each point in the variety has a Voronoi cell whose geometry depends on the normal space at the…
Moving and fixed bottlenecks are moving or fixed capacity restrictions that affect the propagation of traffic flow. They are a very important modeling approach to describe the effects of slow vehicles and traffic signals in transportation…
In this article, we provide new structural results and algorithms for the Homotopy Height problem. In broad terms, this problem quantifies how much a curve on a surface needs to be stretched to sweep continuously between two positions. More…
Tropical varieties capture combinatorial information about how coordinates of points in a classical variety approach zero or infinity. We present algorithms for computing the rays of a complex and real tropical curve defined by polynomials…
Computing the diameter of the intersection graphs of objects is a basic problem in computational geometry. Previous works showed that the complexity of computing the diameter mainly depends on the object types: for unit disks and squares in…
We use a well known problem in discrete and computational geometry (partitions of measures by $k$-fans) as a motivation and as a point of departure to illustrate many aspects, both theoretical and computational, of the problem of…
We investigate a scheme-theoretic variant of Whitney condition a. If X is a projec-tive variety over the field of complex numbers and Y $\subset$ X a subvariety, then X satisfies generically the scheme-theoretic Whitney condition a along Y…
Low-rank approximation of kernels is a fundamental mathematical problem with widespread algorithmic applications. Often the kernel is restricted to an algebraic variety, e.g., in problems involving sparse or low-rank data. We show that…
Many discrete optimization problems amount to selecting a feasible set of edges of least weight. We consider in this paper the context of spatial graphs where the positions of the vertices are uncertain and belong to known uncertainty sets.…