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 c 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: a c d1 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 C 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.
For lack of space, most proofs are omitted. A full version will appear in Combinatorica.
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Babai, L. (1984). On Lovász' lattice reduction and the nearest lattice point problem. In: Mehlhorn, K. (eds) STACS 85. STACS 1985. Lecture Notes in Computer Science, vol 182. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0023990
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DOI: https://doi.org/10.1007/BFb0023990
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