On Lovász’ lattice reduction and the nearest lattice point problem

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 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: ac d1 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.

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Babai, L. On Lovász’ lattice reduction and the nearest lattice point problem. Combinatorica 6, 1–13 (1986). https://doi.org/10.1007/BF02579403

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