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Approximating good simultaneous Diophantine approximations is almost NP-hard

  • Carsten Rössner
  • Jean-Pierre Seifert
Contributed Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1113)

Abstract

Given a real vector α=(α1,..., α d ) and a real number ε>0 a good Diophantine approximation to α is a number Q such that ∥ mod ℤ∥ε, where ∥ · ∥ denotes the ℓ-norm ∥x∥t8 ≔ max1 ≤i≤d ¦ xi¦ for x=(x1,..., xd).

Lagarias [12] proved the NP-completeness of the corresponding decision problem, i.e., given a vector α ∈ ℚ d , a rational number ε>0 and a number Nn+, decide whether there exists a number Q with 1 ≤Q ≤N and ¦| mod ℤ∥t8ε.

We prove that, unless NP⊑DTIME(npoly(log n)), there exists no polynomial-time algorithm which computes on inputs αd and Nn+a number Q* with
and
where γ is an arbitrary small positive constant. To put it in other words, it is almost NP-hard to approximate a minimum good Diophantine approximation to α in polynomial-time within a factor
for an arbitrary small positive constant γ.

We also investigate the nonhomogeneous variant of the good Diophantine approximation problem, i.e., given vectors α, β ∈d, a rational number ε > 0 and a number N ∈n+, decide whether there exists a number Q with 1 ≤Q ≤ N and ∥Qα-β mod ℤ∥t8, ≤ε.

This problem is particularly interesting since finding good nonhomogeneous Diophantine approximations enables us to factor integers and compute discrete logarithms (see Schnorr [17]).

We prove that the problem Good Nonhomogeneous Diophantine Approximation is NP-complete and even approximating it in polynomial-time within a factor
for an arbitrary small positive constant γ is almost NP-hard.

Our results follow from recent work in the theory of probabilistically checkable proofs [4] and 2-prover 1-round interactive proof-systems [7, 14].

Key Words

approximation algorithm computational complexity NP-hard probabilistically checkable proofs Diophantine approximation 2-prover 1-round interactive proof-systems 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Carsten Rössner
    • 1
  • Jean-Pierre Seifert
    • 1
  1. 1.Dept. of Math. Comp. ScienceUniversity of FrankfurtFrankfurt/MainGermany

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