A O(1/eps^2)^n Time Sieving Algorithm for Approximate Integer Programming

TL;DR

This paper introduces an efficient algorithm for approximate integer programming that operates in exponential time relative to the dimension, utilizing a sieving technique adapted for near-symmetric semi-norms.

Contribution

It extends the AKS sieving method to near-symmetric semi-norms and applies it to approximate integer programming and convex bodies.

Findings

Algorithm runs in O(1/ε^2)^n time

Extends sieving techniques to near-symmetric semi-norms

Applicable to general convex bodies and lattices

Abstract

The Integer Programming Problem (IP) for a polytope P \subseteq R^n is to find an integer point in P or decide that P is integer free. We give an algorithm for an approximate version of this problem, which correctly decides whether P contains an integer point or whether a (1+\eps) scaling of P around its barycenter is integer free in time O(1/\eps^2)^n. We reduce this approximate IP question to an approximate Closest Vector Problem (CVP) in a "near-symmetric" semi-norm, which we solve via a 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.