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 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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- 1.L. Adleman, On breaking the iterated Merkle-Hellman public key cryptosystem, Proc. 15th ACM Symp. on Theory of Computing, Boston 1983, 402–412.Google Scholar
- 2.J. W. S. Cassels, An introduction to the geometry of numbers, Springer, New York 1971.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.Google Scholar
- 4.M. Grötschel, L. Lovász and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1 (1981), 186–197.Google Scholar
- 5.M. Grötschel, L. Lovász and A. 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.R. Kannan, A. K. Lenstra and L. 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.Google Scholar
- 7.J. Lagarias and A. M. Odlyzko, Solving low density subset sum problems, in: Proc. 24th IEEE Symp. on Foundations of Comp. Sci., 1983, 1–10.Google Scholar
- 8.A. K. Lenstra, Lattices and factorization of polynomials, Report IW 190/81, Mathematisch Centrum, Amsterdam 1981.Google Scholar
- 9.A. K. Lenstra, H. W. Lenstra, Jr. and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515–534.Google Scholar
- 10.H. W. Lenstra, Jr., Integer programming with a fixed number of variables, Math. Oper. Res. 8 (1983), 538–548.Google Scholar
- 11.L. Lovász, private communications, 1981–1982.Google Scholar
- 12.A. M. Odlyzko and H. te Riele, Disproof of the Mertens conjecture, to appear.Google Scholar
- 13.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.Google Scholar
- 14.Vera T. Sós, On the theory of diophantine approximation II, Acta Math. Acad. Sci. Hung. (1958), 229–241.Google Scholar
- 15.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.Google Scholar