# 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}
}
• Published 1 July 2002
• Mathematics
• Combinatorica
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
• Mathematics
IPCO
• 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
• Mathematics
APPROX-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
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 Science
2008 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.

## References

SHOWING 1-10 OF 16 REFERENCES
An integer analogue of Carathéodory's theorem
• Mathematics
J. Comb. Theory, Ser. B
• 1986
A counterexample to an integer analogue of Carathéodory's theorem
• Mathematics
• 1999
is called an integral polyhedral cone generated by {z1, . . . , zk}. It is called pointed if the origin is a vertex of C and it is called unimodular if the set of generators {z1, . . . , zk} of C
Total Dual Integrality and Integer Polyhedra*
A linear system Ax < b (A, b rational) is said to be totally dual integral (TDI) if for any integer objective function c such that max { cx : Ax <b} exists, there is an integer optimum dual solution.
CONVEX BODIES AND ALGEBRAIC GEOMETRY
During the last decade a new area of research has developed relating two subjects which until now had very little in common: convexity and algebraic geometry. An initial success was Stanley's
On cutting planes
Abstract : The note presents some practical considerations for the implementation of cutting planes of the type known in the literature as convexity or intersection cuts. (Author)
Convex bodies and algebraic geometry, Springer-Verlag
• 1988