Related papers: Exact Line Packings from Numerical Solutions
This paper applies techniques from algebraic and differential geometry to determine how to best pack points in real projective spaces. We present a computer-assisted proof of the optimality of a particular 6-packing in…
How can we arrange $n$ lines through the origin in three-dimensional Euclidean space in a way that maximizes the minimum interior angle between pairs of lines? Conway, Hardin and Sloane (1996) produced line packings for $n \leq 55$ that…
In 2008, Schmidt and Tuller stated a conjecture concerning optimal packing and covering of integers by translates of a given three-point set. In this note, we confirm their conjecture and relate it to several other problems in…
The optimal packings of n unit discs in the plane are known for those natural numbers n, which satisfy certain number theoretic conditions. Their geometric realizations are the extremal Groemer packings (or Wegner packings). But an extremal…
We survey the main formulations and solution methods for two-dimensional orthogonal cutting and packing problems, where both items and bins are rectangles. We focus on exact methods and relaxations for the four main problems from the…
It is often of interest to identify a given number of points in projective space such that the minimum distance between any two points is as large as possible. Such configurations yield representations of data that are optimally robust to…
The construction of optimal line packings in real or complex Euclidean spaces has shown to be a tantalizingly difficult task, because it includes the problem of finding maximal sets of equiangular lines. In the regime where equiangular…
This paper presents theoretical and practical results for the bin packing problem with scenarios, a generalization of the classical bin packing problem which considers the presence of uncertain scenarios, of which only one is realized. For…
Given a family of feasible subsets of a ground set, the packing problem is to find a largest subfamily of pairwise disjoint family members. Non-approximability renders heuristics attractive viable options, while efficient methods with…
We provide a general program for finding nice arrangements of points in real or complex projective space from transitive actions of finite groups. In many cases, these arrangements are optimal in the sense of maximizing the minimum…
Recent LLM-driven discoveries have renewed interest in geometric packing problems. In this paper, we study several classes of such packing problems through the lens of modern global nonlinear optimization. Starting from comparatively direct…
We use group schemes to construct optimal packings of lines through the origin. In this setting, optimal line packings are naturally characterized using representation theory, which in turn leads to a necessary integrality condition for the…
Completion problems, of recovering a point from a set of observed coordinates, are abundant in applications to image reconstruction, phylogenetics, and data science. We consider a completion problem coming from algebraic statistics: to…
We consider the problem of packing rectangles into bins that are unit squares, where the goal is to minimize the number of bins used. All rectangles have to be packed non-overlapping and orthogonal, i.e., axis-parallel. We present an…
A review of a recent method is presented to construct certain exact solutions to the focusing nonlinear Schr\"odinger equation on the line with a cubic nonlinearity. With motivation by the inverse scattering transform and help from the…
We establish the exact overlaps conjecture for iterated functions systems on the real line with algebraic contractions and arbitrary translations.
We consider the problem of packing congruent circles with the maximum radius in a unit square as a mathematical optimization problem. Due to the presence of non-overlapping constraints, this problem is a notoriously difficult nonconvex…
Recently, several intriguing conjectures have been proposed connecting symmetric informationally complete quantum measurements (SIC POVMs, or SICs) and algebraic number theory. These conjectures relate the SICs and their minimal defining…
In this paper we formulate the problem of packing unequal rectangles/squares into a fixed size circular container as a mixed-integer nonlinear program. Here we pack rectangles so as to maximise some objective (e.g. maximise the number of…
An integer program is called ideal if its continuous relaxation coincides with its convex hull allowing the problem to be solved as a continuous program and offering substantial computational advantages. Proving idealness analytically can…