Abstract
The Integer Programming Problem (IP) for a polytope P ⊆ ℝn is to find an integer point in P or decide that P is integer free. We give a randomized algorithm for an approximate version of this problem, which correctly decides whether P contains an integer point or whether a (1 + ε)-scaling of P about its center of gravity is integer free in O(1/ε 2)n-time and O(1/ε)n-space with overwhelming probability. We reduce this approximate IP question to an approximate Closest Vector Problem (CVP) in a “near-symmetric” semi-norm, which we solve via a randomized sieving technique first developed by Ajtai, Kumar, and Sivakumar (STOC 2001). Our main technical contribution is an extension of the AKS sieving technique which works for any near-symmetric semi-norm. Our results also extend to general convex bodies and lattices.
Keywords
- Integer Programming
- Shortest Vector Problem
- Closest Vector Problem
We omit many proofs in this extended abstract. The full version is available at http://arxiv.org/abs/1109.2477.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Ajtai, M., Kumar, R., Sivakumar, D.: A sieve algorithm for the shortest lattice vector problem. In: STOC, pp. 601–610 (2001)
Ajtai, M., Kumar, R., Sivakumar, D.: Sampling short lattice vectors and the closest lattice vector problem. In: IEEE Conference on Computational Complexity, pp. 53–57 (2002)
Arvind, V., Joglekar, P.S.: Some sieving algorithms for lattice problems. In: FSTTCS, pp. 25–36 (2008)
Barvinok, A.: A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed. Mathematics of Operations Research 19(4), 769–779 (1994)
Blömer, J., Naewe, S.: Sampling Methods for Shortest Vectors, Closest Vectors and Successive Minima. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 65–77. Springer, Heidelberg (2007)
Blömer, J., Naewe, S.: Sampling methods for shortest vectors, closest vectors and successive minima. Theoretical Computer Science 110, 1648–1665 (2009)
Dadush, D., Peikert, C., Vempala, S.: Enumerative lattice algorithms in any norm via m-ellipsoid coverings. In: FOCS (2011)
Dyer, M.E., Frieze, A.M., Kannan, R.: A random polynomial time algorithm for approximating the volume of convex bodies. In: STOC, pp. 375–381 (1989)
Eisenbrand, F., Hähnle, N., Niemeier, M.: Covering cubes and the closest vector problem. In: Proceedings of the 27th Annual ACM Symposium on Computational Geometry, SoCG 2011, pp. 417–423. ACM, New York (2011)
Eisenbrand, F., Shmonin, G.: Parametric integer programming in fixed dimension. Mathematics of Operations Research 33(4), 839–850 (2008)
Goldreich, O., Micciancio, D., Safra, S., Seifert, J.P.: Approximating shortest lattice vectors is not harder than approximating closest lattice vectors. Inf. Process. Lett. 71(2), 55–61 (1999)
Gomory, R.: An outline of an algorithm for solving integer programs. Bulletin of the American Mathematical Society 64(5), 275–278 (1958)
Heinz, S.: Complexity of integer quasiconvex polynomial optimization. Journal of Complexity 21(4), 543–556 (2005); festschrift for the 70th Birthday of Arnold Schonhage
Hildebrand, R., Köppe, M.: A new lenstra-type algorithm for quasiconvex polynomial integer minimization with complexity 2O(nlogn). Arxiv, Report 1006.4661 (2010), http://arxiv.org
Kannan, R., Lovász, L., Simonovits, M.: Isoperimetric problems for convex bodies and a localization lemma. Discrete & Computational Geometry 13, 541–559 (1995)
Kannan, R.: Minkowski’s convex body theorem and integer programming. Mathematics of Operations Research 12(3), 415–440 (1987)
Kannan, R.: Test sets for integer programs, ∀ ∃ sentences. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 1, pp. 39–47 (1990)
Lenstra, H.W.: Integer programming with a fixed number of variables. Mathematics of Operations Research 8(4), 538–548 (1983)
Micciancio, D., Voulgaris, P.: A deterministic single exponential time algorithm for most lattice problems based on Voronoi cell computations. In: STOC, pp. 351–358 (2010)
Milman, V., Pajor, A.: Entropy and asymptotic geometry of non-symmetric convex bodies. Advances in Mathematics 152(2), 314–335 (2000)
Paouris, G.: Concentration of mass on isotropic convex bodies. Comptes Rendus Mathematique 342(3), 179–182 (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dadush, D. (2012). A O(1/ε 2)n-Time Sieving Algorithm for Approximate Integer Programming. In: Fernández-Baca, D. (eds) LATIN 2012: Theoretical Informatics. LATIN 2012. Lecture Notes in Computer Science, vol 7256. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29344-3_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-29344-3_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-29343-6
Online ISBN: 978-3-642-29344-3
eBook Packages: Computer ScienceComputer Science (R0)
