Nested Convex Bodies are Chaseable

TL;DR

This paper introduces the first dimension-dependent competitive algorithm for the nested convex body chasing problem, a fundamental challenge in online algorithms, advancing understanding of geometric online decision-making.

Contribution

It provides the first $f(d)$-competitive algorithm specifically for nested convex bodies in $R^d$, addressing a key open problem in online convex body chasing.

Findings

Established the first $f(d)$-competitive algorithm for nested convex bodies.

Connected the nested setting to extending online LP frameworks.

Highlighted the nested setting's role in understanding the general problem.

Abstract

In the Convex Body Chasing problem, we are given an initial point $v_0$ in $R^d$ and an online sequence of $n$ convex bodies $F_1, ..., F_n$. When we receive $F_i$, we are required to move inside $F_i$. Our goal is to minimize the total distance travelled. This fundamental online problem was first studied by Friedman and Linial (DCG 1993). They proved an $\Omega(\sqrt{d})$ lower bound on the competitive ratio, and conjectured that a competitive ratio depending only on d is possible. However, despite much interest in the problem, the conjecture remains wide open. We consider the setting in which the convex bodies are nested: $F_1 \supset ... \supset F_n$. The nested setting is closely related to extending the online LP framework of Buchbinder and Naor (ESA 2005) to arbitrary linear constraints. Moreover, this setting retains much of the difficulty of the general setting and captures an essential obstacle in resolving Friedman and Linial's conjecture. In this work, we give the first $f(d)$-competitive algorithm for chasing nested convex bodies in $R^d$.