# Diophantine Approximations and Integer Points of Cones

@article{Henk2002DiophantineAA, title={Diophantine Approximations and Integer Points of Cones}, author={Martin Henk and Robert Weismantel}, journal={Combinatorica}, year={2002}, volume={22}, pages={401-408} }

The purpose of this note is to present a relation between directed best approximations of a rational vector and the elements of the minimal Hilbert basis of certain rational pointed cones. Furthermore, we show that for a special class of these cones the integer Carathéodory property holds true.

## 6 Citations

Successive Minima and Best Simultaneous Diophantine Approximations

- Mathematics
- 2005

Abstract.We study the problem of best approximations of a vector $\alpha\in{\Bbb R}^n$ by rational vectors of a lattice $\Lambda\subset{\Bbb R}^n$ whose common denominator is bounded. To this end we…

The Mixing Set with Divisible Capacities

- MathematicsIPCO
- 2008

This work studies the special case of divisible capacities, i.e. Ct/Ct-1 is a positive integer for 1 ≥ t ≥ m, and gives an extended formulation for the convex hull of the above set that uses a quadratic number of variables and constraints.

New Hardness Results for Diophantine Approximation

- MathematicsAPPROX-RANDOM
- 2009

It is proved that the mixing set problem with arbitrary capacities is NP-hard, and it is shown that a directed version of Diophantine approximation is also hard to approximate.

On the computational complexity of periodic scheduling

- Computer Science
- 2009

The complexity status of the periodic scheduling problem is settled by proving its NP-hardness, even if one asks for modest approximations, and the more practically oriented area of Real-time scheduling and the field of algorithmic number theory is bridged.

Static-Priority Real-Time Scheduling: Response Time Computation Is NP-Hard

- Computer Science2008 Real-Time Systems Symposium
- 2008

It is shown that the response time of a task cannot be approximated within any constant factor, unless P=NP, which means that response time computation for rate-monotonic, preemptive scheduling of periodic tasks is NP-hard under Turing reductions.

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