Unbounded lower bound for k-server against weak adversaries

TL;DR

This paper proves that for the k-server problem against weak adversaries on general metrics, the competitive ratio cannot be bounded by a constant and grows at least logarithmically with the double logarithm of h, even with infinite servers.

Contribution

It establishes a fundamental lower bound of mega(log h) on the competitive ratio for the problem on general metrics, resolving an open question.

Findings

Lower bound applies to both deterministic and randomized algorithms.

Disproves existence of a competitive algorithm for the infinite server problem on general metrics.

Shows the competitive ratio grows at least logarithmically with double logarithm of h.

Abstract

We study the resource augmented version of the $k$-server problem, also known as the $k$-server problem against weak adversaries or the $(h,k)$-server problem. In this setting, an online algorithm using $k$ servers is compared to an offline algorithm using $h$ servers, where $h\le k$. For uniform metrics, it has been known since the seminal work of Sleator and Tarjan (1985) that for any $\epsilon>0$, the competitive ratio drops to a constant if $k=(1+\epsilon) \cdot h$. This result was later generalized to weighted stars (Young 1994) and trees of bounded depth (Bansal et al. 2017). The main open problem for this setting is whether a similar phenomenon occurs on general metrics. We resolve this question negatively. With a simple recursive construction, we show that the competitive ratio is at least $\Omega(\log \log h)$, even as $k\to\infty$. Our lower bound holds for both deterministic and randomized algorithms. It also disproves the existence of a competitive algorithm for the infinite server problem on general metrics.