On Lovász’ lattice reduction and the nearest lattice point problem
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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 ofcd (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: ac1d lower bound on the angle between any member of the basis and the hyperplane generated by the other members, wherec1 = √2/3.
As an application, we obtain a solution to the nonhomogeneous simultaneous diophantine approximation problem, optimal within a factor ofCd.
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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