A Nearly-Linear Bound for Chasing Nested Convex Bodies

TL;DR

This paper presents an $O(d \, \log d)$-competitive algorithm for the nested convex body chasing problem in any norm, significantly improving previous bounds and nearly matching the known lower bounds.

Contribution

The authors introduce a new strategy for nested convex body chasing that achieves near-optimal competitiveness, advancing understanding in geometric online algorithms.

Findings

Achieves $O(d \log d)$ competitiveness for nested convex body chasing.

Works for any norm, broadening applicability.

Nearly matches the known $\Omega(d)$ lower bound.

Abstract

Friedman and Linial introduced the convex body chasing problem to explore the interplay between geometry and competitive ratio in metrical task systems. In convex body chasing, at each time step $t \in \mathbb{N}$, the online algorithm receives a request in the form of a convex body $K_t \subseteq \mathbb{R}^d$ and must output a point $x_t \in K_t$. The goal is to minimize the total movement between consecutive output points, where the distance is measured in some given norm. This problem is still far from being understood, and recently Bansal et al. gave an algorithm for the nested version, where each convex body is contained within the previous one. We propose a different strategy which is $O(d \log d)$-competitive algorithm for this nested convex body chasing problem, improving substantially over previous work. Our algorithm works for any norm. This result is almost tight, given an $\Omega(d)$ lower bound for the $\ell_{\infty}$.