Near-Optimal Streaming Ellipsoidal Rounding for General Convex Polytopes

TL;DR

This paper presents near-optimal streaming algorithms for ellipsoidal rounding of general convex polytopes, achieving near-matching runtime and approximation bounds, and extends to coreset construction for convex hulls.

Contribution

It introduces the first streaming algorithms for ellipsoidal rounding of general convex polytopes with near-optimal approximation and runtime, surpassing prior symmetric-only methods.

Findings

Algorithms nearly match offline runtimes.

Approximation factors are close to theoretical lower bounds.

Applicable to general convex polytopes, not just symmetric ones.

Abstract

We give near-optimal algorithms for computing an ellipsoidal rounding of a convex polytope whose vertices are given in a stream. The approximation factor is linear in the dimension (as in John's theorem) and only loses an excess logarithmic factor in the aspect ratio of the polytope. Our algorithms are nearly optimal in two senses: first, their runtimes nearly match those of the most efficient known algorithms for the offline version of the problem. Second, their approximation factors nearly match a lower bound we show against a natural class of geometric streaming algorithms. In contrast to existing works in the streaming setting that compute ellipsoidal roundings only for centrally symmetric convex polytopes, our algorithms apply to general convex polytopes. We also show how to use our algorithms to construct coresets from a stream of points that approximately preserve both the ellipsoidal rounding and the convex hull of the original set of points.