Diophantine Approximations and Integer Points of Cones

  title={Diophantine Approximations and Integer Points of Cones},
  author={Martin Henk and Robert Weismantel},
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. 
Successive Minima and Best Simultaneous Diophantine Approximations
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
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New Hardness Results for Diophantine Approximation
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.
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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
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.


A counterexample to an integer analogue of Carathéodory's theorem
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
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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.
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
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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