Nicolai Hähnle

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We investigate the diameter of a natural abstraction of the 1-skeleton of polyhedra. Although this abstraction is simpler than other abstractions that were previously studied in the literature, the best upper bounds on the diameter of polyhedra continue to hold here. On the other hand, we show that this abstraction has its limits by providing a superlinear(More)
We study the simplex method over polyhedra satisfying certain “discrete curvature” lower bounds, which enforce that the boundary always meets vertices at sharp angles. Motivated by linear programs with totally unimodular constraint matrices, recent results of Bonifas et al. (Discrete Comput. Geom. 52(1):102–115, 2014), Brunsch and Röglin (Automata,(More)
We consider a real-time scheduling problem that occurs in the design of software-based aircraft control. The goal is to distribute tasks τi = (ci, pi) on a minimum number of identical machines and to compute offsets ai for the tasks such that no collision occurs. A task τi releases a job of running time ci at each time ai +k · pi, k ∈N0 and a collision(More)
We derive a new upper bound on the diameter of the graph of a polyhedron P = {x &#8712; R<sup>n</sup> : Ax &#8804; b}, where A &#8712; Z<sup>m&#215;n</sup>. The bound is polynomial in n and the largest absolute value of a sub-determinant of A, denoted by &#916;. More precisely, we show that the diameter of P is bounded by O(&#916;<sup>2</sup> n<sup>4</sup>(More)
We consider the problem of testing whether the maximum additive integrality gap of a family of integer programs in standard form is bounded by a given constant. This can be viewed as a generalization of the integer rounding property, which can be tested in polynomial time if the number of constraints is fixed. It turns out that this generalization is(More)
We provide the currently fastest randomized (1+epsilon)-approximation algorithm for the closest lattice vector problem in the infinity-norm. The running time of our method depends on the dimension n and the approximation guarantee epsilon by 2<sup>(O(n))</sup> (log(1/epsilon))<sup>(O(n))</sup> which improves upon the (2+1/epsilon)<sup>(O(n))</sup> running(More)
Polytopes Abstract Polyhedra Ultraconnected Families Abstract Polyhedral Graphs Blueprints It is unclear which of the inequalities shown in this diagram are strict. What we do know for a fact is that for dimension d ≤ 5, the maximum diameter of APGs is strictly larger than the maximum diameter of abstract polytopes (this follows from [AD74] and theorem(More)
The parametric lattice-point counting problem is as follows: Given an integer matrix A ∈ Zm×n , compute an explicit formula parameterized by b ∈ R that determines the number of integer points in the polyhedron {x ∈R : Ax É b}. In the last decade, this counting problem has received considerable attention in the literature. Several variants of Barvinok’s(More)