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A O(1/ɛ 2) n-time Sieving Algorithm for Approximate Integer Programming

By Daniel Dadush


The Integer Programming Problem (IP) for a polytope P ⊆ R n 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 + ɛ) scaling of P around its barycenter is integer free in time O(1/ɛ 2) n 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

Topics: Shortest Vector Problem
Year: 2011
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