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

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+ ǫ) scaling of P around its barycenter is integer free in time O(1/ǫ) 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.