A Deterministic Polynomial Space Construction for eps-nets under any Norm

TL;DR

This paper presents a deterministic polynomial space algorithm for constructing nearly optimal eps-nets for convex bodies under any norm, improving efficiency and space complexity over previous methods.

Contribution

It introduces a novel deterministic polynomial space construction of thin lattice coverings for convex bodies, enabling efficient enumeration and eps-net construction in any norm.

Findings

Achieves 2^O(n) time and polynomial space construction of eps-nets for convex bodies.

Provides the first deterministic polynomial space construction of thin lattice coverings for general convex bodies.

Develops a volume approximation algorithm with nearly optimal dependence on epsilon.

Abstract

We give a deterministic polynomial space construction for nearly optimal eps-nets with respect to any input n-dimensional convex body K and norm |.|. More precisely, our algorithm can build and iterate over an eps-net of K with respect to |.| in time 2^O(n) x (size of the optimal net) using only poly(n)-space. This improves on previous constructions of Alon et al [STOC 2013] which achieve either a 2^O(n) approximation or an n^O(n) approximation of the optimal net size using 2^n space and poly(n)-space respectively. As in their work, our algorithm relies on the mathematically classical approach of building thin lattice coverings of space, which reduces the task of constructing eps-nets to the problem of enumerating lattice points. Our main technical contribution is a deterministic 2^O(n)-time and poly(n)-space construction of thin lattice coverings of space with respect to any convex body, where enumeration in these lattices can be efficiently performed using poly(n)-space. This also yields the first existential construction of poly(n)-space enumerable thin covering lattices for general convex bodies, which we believe is of independent interest. Our construction combines the use of the M-ellipsoid from convex geometry with lattice sparsification and densification techniques. As an application, we give a 2^O(n)(1+1/eps)^n time and poly(n)-space deterministic algorithm for computing a (1+eps)^n approximation to the volume of a general convex body, which nearly matches the lower bounds for volume estimation in the oracle model (the dependence on eps is larger by a factor 2 in the exponent). This improves on the previous results of Dadush and Vempala [PNAS 2013], which gave the above result only for symmetric bodies and achieved a dependence on eps of (1+log^{5/2}(1/eps)/eps^2)^n.