@misc{BeckSchymura, author = {Beck, Matthias and Schymura, Matthias}, title = {Deep Lattice Points in Zonotopes, Lonely Runners, and Lonely Rabbits}, series = {International Mathematics Research Notices}, volume = {2024}, journal = {International Mathematics Research Notices}, number = {8}, publisher = {Oxford University Press (OUP)}, issn = {1073-7928}, doi = {https://doi.org/10.1093/imrn/rnad232}, pages = {6553 -- 6578}, language = {en} } @misc{AverkovGonzalezMerinoPaschkeetal., author = {Averkov, Gennadiy and Gonz{\´a}lez Merino, Bernardo and Paschke, Ingo and Schymura, Matthias and Weltge, Stefan}, title = {Tight bounds on discrete quantitative Helly numbers}, series = {Advances in Applied Mathematics}, volume = {89}, journal = {Advances in Applied Mathematics}, issn = {0196-8858}, doi = {10.1016/j.aam.2017.04.003}, pages = {76 -- 101}, language = {en} } @misc{SchymuraSeidelWeltge, author = {Schymura, Matthias and Seidel, Ina and Weltge, Stefan}, title = {Lifts for Voronoi cells of lattices}, series = {Discrete \& Computational Geometry}, volume = {70}, journal = {Discrete \& Computational Geometry}, number = {3}, publisher = {Springer US}, issn = {0179-5376}, doi = {10.1007/s00454-023-00522-z}, pages = {845 -- 865}, abstract = {Many polytopes arising in polyhedral combinatorics are linear projections of higher-dimensional polytopes with significantly fewer facets. Such lifts may yield compressed representations of polytopes, which are typically used to construct small-size linear programs. Motivated by algorithmic implications for the closest vector problem, we study lifts of Voronoi cells of lattices. We construct an explicit d -dimensional lattice such that every lift of the respective Voronoi cell has 2Ω(d/logd)facets. On the positive side, we show that Voronoi cells of d -dimensional root lattices and their dual lattices have lifts with O(d)and O(dlogd)facets, respectively. We obtain similar results for spectrahedral lifts.}, language = {en} } @misc{CodenottiSantosSchymura, author = {Codenotti, Giulia and Santos, Francisco and Schymura, Matthias}, title = {The Covering Radius and a Discrete Surface Area for Non-Hollow Simplices}, series = {Discrete \& Computational Geometry}, volume = {67}, journal = {Discrete \& Computational Geometry}, number = {1}, issn = {0179-5376}, doi = {10.1007/s00454-021-00330-3}, pages = {65 -- 111}, language = {en} } @misc{AverkovHojnySchymura, author = {Averkov, Gennadiy and Hojny, Christopher and Schymura, Matthias}, title = {Computational aspects of relaxation complexity: possibilities and limitations}, series = {Mathematical Programming}, volume = {197}, journal = {Mathematical Programming}, number = {2}, issn = {0025-5610}, doi = {10.1007/s10107-021-01754-8}, pages = {1173 -- 1200}, language = {en} } @misc{AverkovSchymura, author = {Averkov, Gennadiy and Schymura, Matthias}, title = {Complexity of linear relaxations in integer programming}, series = {Mathematical Programming}, volume = {194}, journal = {Mathematical Programming}, number = {1-2}, issn = {0025-5610}, doi = {10.1007/s10107-021-01623-4}, pages = {191 -- 227}, language = {en} } @misc{AverkovSchymura, author = {Averkov, Gennadiy and Schymura, Matthias}, title = {On the maximal number of columns of a Delta-modular integer matrix: bounds and computations}, series = {Mathematical Programming, Series B}, volume = {2023}, journal = {Mathematical Programming, Series B}, issn = {1436-4646}, doi = {10.1007/s10107-023-01964-2}, pages = {29}, language = {en} } @misc{AverkovHojnySchymura, author = {Averkov, Gennadiy and Hojny, Christopher and Schymura, Matthias}, title = {Efficient MIP techniques for computing relaxation complexity}, series = {Mathematical Programming Computation}, volume = {15}, journal = {Mathematical Programming Computation}, number = {3}, issn = {1867-2957}, doi = {10.1007/s12532-023-00241-9}, pages = {549 -- 580}, language = {en} } @misc{HenkSchymuraXue, author = {Henk, Martin and Schymura, Matthias and Xue, Fei}, title = {Packing minima and lattice points in convex bodies}, series = {Moscow Journal of Combinatorics and Number Theory}, volume = {10}, journal = {Moscow Journal of Combinatorics and Number Theory}, number = {1}, publisher = {Mathematical Sciences Publishers}, issn = {2640-7361}, doi = {10.2140/moscow.2021.10.25}, pages = {25 -- 48}, abstract = {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 packing minima, associated to a convex body K and a lattice Λ. These numbers interpolate between the successive minima of K and the inverse of the successive minima of the polar body of K and can be understood as packing counterparts to the covering minima of Kannan \& Lov{\´a}sz (1988). As our main results, we prove sharp inequalities that relate the volume and the number of lattice points in K to the sequence of packing minima. Moreover, we extend classical transference bounds and discuss a natural class of examples in detail.}, language = {en} } @misc{CslovjecsekMalikiosisNaszodietal., author = {Cslovjecsek, Jana and Malikiosis, Romanos Diogenes and Nasz{\´o}di, M{\´a}rton and Schymura, Matthias}, title = {Computing the Covering Radius of a Polytope with an Application to Lonely Runners}, series = {Combinatorica}, volume = {42}, journal = {Combinatorica}, number = {4}, publisher = {Springer Science and Business Media LLC}, issn = {0209-9683}, doi = {10.1007/s00493-020-4633-8}, pages = {463 -- 490}, abstract = {We study the computational problem of determining the covering radius of a rational polytope. This parameter is defined as the minimal dilation factor that is needed for the lattice translates of the correspondingly dilated polytope to cover the whole space. As our main result, we describe a new algorithm for this problem, which is simpler, more efficient and easier to implement than the only prior algorithm of Kannan (1992). Motivated by a variant of the famous Lonely Runner Conjecture, we use its geometric interpretation in terms of covering radii of zonotopes, and apply our algorithm to prove the first open case of three runners with individual starting points.}, language = {en} }