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A O(1/ε 2)n-Time Sieving Algorithm for Approximate Integer Programming

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7256)

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.

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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

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  • 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

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