Minkowski's convex body theorem and integer programming
Mathematics of Operations Research
A random polynomial time algorithm for approximating the volume of convex bodies
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed
Mathematics of Operations Research
Approximating shortest lattice vectors is not harder than approximating closet lattice vectors
Information Processing Letters
A sieve algorithm for the shortest lattice vector problem
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Sampling Short Lattice Vectors and the Closest Lattice Vector Problem
CCC '02 Proceedings of the 17th IEEE Annual Conference on Computational Complexity
Sampling methods for shortest vectors, closest vectors and successive minima
Theoretical Computer Science
Complexity of integer quasiconvex polynomial optimization
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
Proceedings of the forty-second ACM symposium on Theory of computing
Covering cubes and the closest vector problem
Proceedings of the twenty-seventh annual symposium on Computational geometry
Enumerative Lattice Algorithms in any Norm Via M-ellipsoid Coverings
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Sampling methods for shortest vectors, closest vectors and successive minima
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
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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/ε2)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.