We derive a new upper bound on the diameter of a polyhedron P = {x ∈ R n : Ax b}, where A ∈ Z m×n . The bound is polynomial in n and the largest absolute value of a sub-determinant of A, denoted by ∆. More precisely, we show that the diameter of P is bounded by O ∆ 2 n 4 log n∆ . If P is bounded, then we show that the diameter of P is at most O ∆ 2 n 3.5 log n∆ .For the special case in which A is a totally unimodular matrix, the bounds are O n 4 log n and O n 3.5 log n respectively. This improves over the previous best bound of O(m 16 n 3 (log mn) 3 ) due to Dyer and Frieze [DF94]. * An extended abstract of this paper was presented at the 28-th annual ACM symposium on Computational Geometry (SOCG 12) † LIX, École Polytechnique, Palaiseau and IBM,

We provide the currently fastest randomized (1 + ε)-approximation algorithm for the closest lattice vector problem in the ℓ∞-norm. The running time of our method depends on the dimension n and the approximation guarantee ε bywhich improves upon the (2 + 1/ε) O(n) running time of the previously best algorithm by Blömer and Naewe.Our algorithm is based on a solution of the following geometric covering problem that is of interest of its own: Given ε ∈ (0, 1), how many ellipsoids are necessary to cover the cube [−1 + ε, 1 − ε] n such that all ellipsoids are contained in the standard unit cube [−1, 1] n ? We provide an almost optimal bound for the case where the ellipsoids are restricted to be axis-parallel.We then apply our covering scheme to a variation of this covering problem where one wants to cover [−1 + ε, 1 − ε] n with parallelepipeds that, if scaled by two, are still contained in the unit cube. Thereby, we obtain a method to boost any 2-approximation algorithm for closest-vector in the ℓ∞-norm to a (1 + ε)-approximation algorithm that has the desired running time.

Abstract. We give a rigorous account on the complexity landscape of an important real-time scheduling problem that occurs in the design of software-based aircraft control. The goal is to distribute tasks τ i = (c i , p i ) on a minimum number of identical machines and to compute offsets a i for the tasks such that no collision occurs. A task τ i releases a job of running time c i at each time a i + k · p i , k ∈ N 0 and a collision occurs if two jobs are simultaneously active on the same machine. Our main results are as follows: (i) We show that the minimization problem cannot be approximated within a factor of n 1−ε for any ε > 0. (ii) If the periods are harmonic (for each i, j one has p i | p j or p j | p i ), then there exists a 2-approximation for the minimization problem and this result is tight, even asymptotically. (iii) We provide asymptotic approximation schemes in the harmonic case if the number of different periods is constant.

We report on the solution of a real-time scheduling problem that arises in the design of software-based operation control of aircraft. A set of tasks has to be distributed on a minimum number of machines and offsets of the tasks have to be computed. The tasks emit jobs periodically starting at their offset and then need to be executed on the machines without any delay. Also, further constraints in terms of memory usage and redundancy requirements have to be met. Approaches based on standard integer programming formulations fail to solve our real-world instances. By exploiting structural insights of the problem we obtain an IP-formulation and primal heuristics that together solve the real-world instances to optimality and outperform text-book approaches by several orders of magnitude. Our methods lead, for the first time, to an industry strength tool to optimally schedule aircraft sized problems.

Abstract. We address a scheduling problem that arises in highly parallelized environments like modern multi-core CPU/GPU computer architectures. Here simultaneously active jobs share a common limited resource, e.g., memory cache. The scheduler must ensure that the demand for the common resource never exceeds the available capacity. This introduces an orthogonal constraint to the classical minimum makespan scheduling problem. Such a constraint also arises in many other contexts where a common resource is shared across the machines. We study the non-preemptive case of this problem and give a (2 + )-approximation algorithm which relies on the interplay of several classical and modern techniques in scheduling like grouping, job-classification, and the use of configuration-LPs. This improves upon previous bound of 3 that can be obtained by list scheduling approaches, and gets close to the (3/2 − ) inapproximability bound. If the number of machines or the number of different resource requirements are bounded by a constant we have a polynomial time approximation scheme.