Enumerative Lattice Algorithms in Any Norm via M-Ellipsoid Coverings

TL;DR

This paper introduces a new lattice point enumeration algorithm applicable in any norm using M-ellipsoid coverings, improving deterministic solutions for classic lattice problems and integer programming with applications across various convex geometries.

Contribution

It presents a novel enumeration technique based on M-ellipsoids, including a deterministic polynomial-time algorithm for computing M-ellipsoids in many norms, and derandomizes existing lattice problem algorithms.

Findings

Deterministic 2^{O(n)}-time algorithms for SVP and CVP in any norm with an M-ellipsoid.

An expected O(f*(n))^n-time algorithm for Integer Programming, improving previous bounds.

A new method for computing M-ellipsoids in polynomial time for many norms, enabling broader applicability.

Abstract

We give a novel algorithm for enumerating lattice points in any convex body, and give applications to several classic lattice problems, including the Shortest and Closest Vector Problems (SVP and CVP, respectively) and Integer Programming (IP). Our enumeration technique relies on a classical concept from asymptotic convex geometry known as the M-ellipsoid, and uses as a crucial subroutine the recent algorithm of Micciancio and Voulgaris (STOC 2010) for lattice problems in the l_2 norm. As a main technical contribution, which may be of independent interest, we build on the techniques of Klartag (Geometric and Functional Analysis, 2006) to give an expected 2^O(n)-time algorithm for computing an M-ellipsoid for any n-dimensional convex body. As applications, we give deterministic 2^{O(n)}-time and -space algorithms for solving exact SVP, and exact CVP when the target point is sufficiently close to the lattice, on n-dimensional lattices in any (semi-)norm given an M-ellipsoid of the unit ball. In many norms of interest, including all l_p norms, an M-ellipsoid is computable in deterministic poly(n) time, in which case these algorithms are fully deterministic. Here our approach may be seen as a derandomization of the "AKS sieve" for exact SVP and CVP (Ajtai, Kumar, and Sivakumar; STOC 2001 and CCC 2002). As a further application of our SVP algorithm, we derive an expected O(f*(n))^n-time algorithm for Integer Programming, where f*(n) denotes the optimal bound in the so-called "flatness theorem," which satisfies f*(n) = O(n^{4/3} \polylog(n)) and is conjectured to be f*(n)=\Theta(n). Our runtime improves upon the previous best of O(n^{2})^{n} by Hildebrand and Koppe (2010).