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Combinatorica

, Volume 6, Issue 1, pp 1–13 | Cite as

On Lovász’ lattice reduction and the nearest lattice point problem

  • L. Babai
Article

Abstract

Answering a question of Vera Sós, we show how Lovász’ lattice reduction can be used to find a point of a given lattice, nearest within a factor ofc d (c = const.) to a given point in R d . We prove that each of two straightforward fast heuristic procedures achieves this goal when applied to a lattice given by a Lovász-reduced basis. The verification of one of them requires proving a geometric feature of Lovász-reduced bases: ac 1 d lower bound on the angle between any member of the basis and the hyperplane generated by the other members, wherec 1 = √2/3.

As an application, we obtain a solution to the nonhomogeneous simultaneous diophantine approximation problem, optimal within a factor ofC d .

In another application, we improve the Grötschel-Lovász-Schrijver version of H. W. Lenstra’s integer linear programming algorithm.

The algorithms, when applied to rational input vectors, run in polynomial time.

AMS subject classification (1980)

68 C 25 10 F 10 10 F 15 90 C 10 

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Copyright information

© Akadémiai Kiadó 1986

Authors and Affiliations

  • L. Babai
    • 1
    • 2
  1. 1.Department of AlgebraEötvös UniversityBudapestHungary
  2. 2.Department of Computer ScienceThe University of ChicagoChicago

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