Related papers: Packings in real projective spaces
We consider a facility location problem, where the objective is to ``disperse'' a number of facilities, i.e., select a given number k of locations from a discrete set of n candidates, such that the average distance between selected…
We describe algorithms which address two classical problems in lattice geometry: the lattice covering and the simultaneous lattice packing-covering problem. Theoretically our algorithms solve the two problems in any fixed dimension d in the…
We present a first exact study on higher-dimensional packing problems with order constraints. Problems of this type occur naturally in applications such as logistics or computer architecture and can be interpreted as higher-dimensional…
We determine putative optimal packings of regular spherical polygons via optimization on smooth manifolds. For several cases, we establish maximality by extending the Lov\'asz theta number to Cayley graphs on the special orthogonal group…
Optimal geometrical arrangements, such as the stacking of atoms, are of relevance in diverse disciplines. A classic problem is the determination of the optimal arrangement of spheres in three dimensions in order to achieve the highest…
Given a polygon $P$, for two points $s$ and $t$ contained in the polygon, their \emph{geodesic distance} is the length of the shortest $st$-path within $P$. A \emph{geodesic disk} of radius $r$ centered at a point $v \in P$ is the set of…
The ability to jointly understand the geometry of objects and plan actions for manipulating them is crucial for intelligent agents. We refer to this ability as geometric planning. Recently, many interactive environments have been proposed…
Packing problems in discrete geometry can be modeled as finding independent sets in infinite graphs where one is interested in independent sets which are as large as possible. For finite graphs one popular way to compute upper bounds for…
In rectangle packing problems we are given the task of placing axis-aligned rectangles in a given plane region, so that they do not overlap with each other. In Maximum Weight Independent Set of Rectangles (MWISR), their position is given…
This paper presents a multiscale approach to efficiently compute approximate optimal transport plans between point sets. It is particularly well-suited for point sets that are in high-dimensions, but are close to being intrinsically…
We consider the problem of finding all enclosing rectangles of minimum area that can contain a given set of rectangles without overlap. Our rectangle packer chooses the x-coordinates of all the rectangles before any of the y-coordinates. We…
We consider packings of congruent circles on a square flat torus, i.e., periodic (w.r.t. a square lattice) planar circle packings, with the maximal circle radius. This problem is interesting due to a practical reason - the problem of "super…
We give a polynomial-time constant-factor approximation algorithm for maximum independent set for (axis-aligned) rectangles in the plane. Using a polynomial-time algorithm, the best approximation factor previously known is $O(\log\log n)$.…
We present filling as a new type of spatial subdivision problem that is related to covering and packing. Filling addresses the optimal placement of overlapping objects lying entirely inside an arbitrary shape so as to cover the most…
In the \textsc{2-Dimensional Knapsack} problem (2DK) we are given a square knapsack and a collection of $n$ rectangular items with integer sizes and profits. Our goal is to find the most profitable subset of items that can be packed…
The problem of packing equal circles in a circle is a classic and famous packing problem, which is well-studied in academia and has a variety of applications in industry. This problem is computationally challenging, and researchers mainly…
Different types of two- and three-dimensional representations of a finite metric space are studied that focus on the accurate representation of the linear order among the distances rather than their actual values. Lower and upper bounds for…
The problem of packing equal spheres in a spherical container is a classic global optimization problem, which has attracted enormous studies in academia and found various applications in industry. This problem is computationally…
Cutting and packing problems are present in many, at first glance unconnected, areas, therefore it's beneficial to have a good understanding of their underlying structure, to select proper techniques for finding solutions. Cutting and…
We give an approximation algorithm for packing and covering linear programs (linear programs with non-negative coefficients). Given a constraint matrix with n non-zeros, r rows, and c columns, the algorithm computes feasible primal and dual…