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On Lovász’ lattice reduction and the nearest lattice point problem


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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  1. L. Adleman, On breaking the iterated Merkle—Hellman public key cryptosystem,Proc. 15th ACM Symp. on Theory of Computing, Boston (1983), 402–412.

  2. J. W. S. Cassels,An introduction to the geometry of numbers, Springer, New York, (1971).

    MATH  Google Scholar 

  3. P. van Emde Boas, Another NP-complete partition problem and the complexity of computing short vectors in a lattice,Rep. MI/UVA 81-04, Amsterdam (1981).

  4. M. Grötschel, L. Lovász andA. Schrijver, The ellipsoid method and its consequences in combinatorial optimization,Combinatorica 1 (1981), 186–197.

    Google Scholar 

  5. M. Grötschel, L. Lovász andA. Schrijver, Geometric methods in combinatorial optimization, in: Progress in Combinatorial Optimization (W. R. Pulleyblank, ed.),Proc. Silver Jubilee Conf. on Comb., Univ. Waterloo, Vol. 1, 1982, Acad. Press, N. Y. (1984), 167–183.

    Google Scholar 

  6. B. Helfrich, An algorithm to construct Minkowski-reduced lattice-bases, in:Proc. 2nd Ann. Symp. on Theoretical Aspects of Comp. Sci. (STACS 85),Springer Lect. Notes. in Comp. Sci. 182 (1985), 173–179.

    MathSciNet  Article  Google Scholar 

  7. R. Kannan, Improved algorithms for integer programming and related lattice problems,in: Proc. 15th ACM Symp. on Theory of Comp., (1983), 193–206.

  8. R. Kannan, A. K. Lenstra andL. Lovász, Polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers,in: Proc. 16th Ann. ACM Symp. on Theory of Computing, Washington, D. C. (1984), 191–200.

  9. J. Lagarias andA. M. Odlyzko, Solving low density subset sum problems,in: Proc. 24th IEEE Symp. on Foundations of Comp. Sci., (1983), 1–10.

  10. A. K. Lenstra, Lattices and factorization of polynomials,Report IW 190/81, Mathematisch Centrum, Amsterdam (1981).

  11. A. K. Lenstra, H. W. Lenstra, Jr. andL. Lovász, Factoring polynomials with rational coefficients,Math. Ann. 261 (1982), 515–534.

    MATH  Article  MathSciNet  Google Scholar 

  12. H. W. Lenstra, Jr., Integer programming with a fixed number of variables.Math. Oper. Res. 8 (1983), 538–548.

    MATH  MathSciNet  Article  Google Scholar 

  13. L. Lovász,private communications, 1981–1982.

  14. A. M. Odlyzko andH. te Riele, Disproof of the Mertens conjecture,J. reine angew. Math. 357 (1985), 138–160.

    MATH  MathSciNet  Google Scholar 

  15. A. Shamir, A polynomial time algorithm for breaking the Merkle—Hellman cryptosystem,Proc. 23rd IEEE Symp. on Foundations of Comp. Sci., Chicago, Illinois (1982), 145–152.

  16. C. P. Schnorr, A hierarchy of polynomial time basis reduction algorithms,in: Theory of Algorithms, Proc. Conf. Pécs (Hungary) 1984,Coll. Soc. J. Bolyai, to appear.

  17. Vera T. Sós, On the theory of diophantine approximation II,Acta Math. Acad. Sci. Hung. (1958), 229–241.

  18. Vera T. Sós, Irregularities of partitions: Ramsey theory, uniform distribution,in: Surveys in Combinatorics, Proc. 9th British Combinatorial Conference, 1983 (E. Keith Lloyd, ed.) London Math. Soc. Lect. Notes 82, Cambridge Univ. Press 1983.

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Babai, L. On Lovász’ lattice reduction and the nearest lattice point problem. Combinatorica 6, 1–13 (1986).

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AMS subject classification (1980)

  • 68 C 25
  • 10 F 10
  • 10 F 15
  • 90 C 10