3 citations · 11 across the 13 of their papers we have counts for
16 papers
On lattice coverings by locally anti-blocking bodies and polytopes with few vertices
Matthias Schymura, Jun Wang, Fei Xue
In 2021, Ordentlich, Regev and Weiss made a breakthrough that the lattice covering density of any -dimensional convex body is upper bounded by , improving on the best pr…
Linearly-exponential checking is enough for the Lonely Runner Conjecture and some of its variants
Romanos Diogenes Malikiosis, Francisco Santos, Matthias Schymura
Tao (2018) showed that in order to prove the Lonely Runner Conjecture (LRC) up to runners it suffices to consider positive integer velocities in the order of . Us…
Deep lattice points in zonotopes, lonely runners, and lonely rabbits
Matthias Beck, Matthias Schymura
Let be a convex body and let be an interior point of . The coefficient of asymmetry $\operatorname{ca}(K,\mathb…
Efficient MIP Techniques for Computing the Relaxation Complexity
Gennadiy Averkov, Christopher Hojny, Matthias Schymura
The relaxation complexity rc(X) of the set of integer points X contained in a polyhedron is the minimal number of inequalities needed to formulate a linear optimization problem ove…
Lifts for Voronoi cells of lattices
Matthias Schymura, Ina Seidel, Stefan Weltge
Many polytopes arising in polyhedral combinatorics are linear projections of higher-dimensional polytopes with significantly fewer facets. Such lifts may yield compressed represent…
Computational Aspects of Relaxation Complexity: Possibilities and Limitations
Gennadiy Averkov, Christopher Hojny, Matthias Schymura
The relaxation complexity of the set of integer points contained in a polyhedron is the smallest number of facets of any polyhedron such that the integer p…
Computing the covering radius of a polytope with an application to lonely runners
Jana Cslovjecsek, Romanos Diogenes Malikiosis, Márton Naszódi +1
We are concerned with the computational problem of determining the covering radius of a rational polytope. This parameter is defined as the minimal dilation factor that is needed f…
Packing minima and lattice points in convex bodies
Martin Henk, Matthias Schymura, Fei Xue
Motivated by long-standing conjectures on the discretization of classical inequalities in the Geometry of Numbers, we investigate a new set of parameters, which we call \emph{packi…
Complexity of linear relaxations in integer programming
Gennadiy Averkov, Matthias Schymura
For a set of integer points in a polyhedron, the smallest number of facets of any polyhedron whose set of integer points coincides with is called the relaxation complexity…
Tropical Ehrhart Theory and Tropical Volume
Georg Loho, Matthias Schymura
We introduce a novel intrinsic volume concept in tropical geometry. This is achieved by developing the foundations of a tropical analog of lattice point counting in polytopes. We e…
The covering radius and a discrete surface area for non-hollow simplices
Giulia Codenotti, Francisco Santos, Matthias Schymura
We explore upper bounds on the covering radius of non-hollow lattice polytopes. In particular, we conjecture a general upper bound of in dimension , achieved by the "stand…
On compact representations of Voronoi cells of lattices
Christoph Hunkenschröder, Gina Reuland, Matthias Schymura
In a seminal work, Micciancio & Voulgaris (2013) described a deterministic single-exponential time algorithm for the Closest Vector Problem (CVP) on lattices. It is based on the co…
On the reverse isodiametric problem and Dvoretzky-Rogers-type volume bounds
Bernardo González Merino, Matthias Schymura
The isodiametric inequality states that the Euclidean ball maximizes the volume among all convex bodies of a given diameter. We are motivated by a conjecture of Makai Jr.~on the re…
A note on discrete lattice-periodic sets with an application to Archimedean tilings
Matthias Schymura, Liping Yuan
Cao & Yuan obtained a Blichfeldt-type result for the vertex set of the edge-to-edge tiling of the plane by regular hexagons. Observing that every Archimedean tiling is the union of…
Lonely Runner Polyhedra
Matthias Beck, Serkan Hosten, Matthias Schymura
We study the \emph{Lonely Runner Conjecture}, conceived by Jörg M.~Wills in the 1960's: Given positive integers , there exists a positive real number such…
On densities of lattice arrangements intersecting every i-dimensional affine subspace
Bernardo González Merino, Matthias Henze
In 1978, Makai Jr. established a remarkable connection between the volume-product of a convex body, its maximal lattice packing density and the minimal density of a lattice arrange…