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On Lovász' lattice reduction and the nearest lattice point problem

Shortened version
  • László Babai
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 182)

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 of cd (c = const.) to a given point in Rd. 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: a c 1 d lower bound on the angle between any member of the basis and the hyperplane generated by the other members, where \(c_1 = \sqrt 2 /3\).

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

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.

For lack of space, most proofs are omitted. A full version will appear in Combinatorica.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • László Babai
    • 1
  1. 1.Department of AlgebraEötvös UniversityBudapestHungary

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