Efficient Algorithms and Hardness Results for the Weighted $k$-Server Problem

TL;DR

This paper investigates the computational complexity and algorithmic solutions for the weighted $k$-server problem on uniform metrics, revealing hardness results and proposing approximation algorithms with resource augmentation in both offline and online settings.

Contribution

It establishes strong hardness bounds assuming the unique games conjecture and introduces a constant-approximation LP rounding algorithm with resource augmentation.

Findings

No polynomial-time approximation within sub-polynomial factor assuming UGC

LP relaxation requires at least $ ext{l}$ resource augmentation for bounded integrality gap

Online algorithm reduces competitive ratio to $O( ext{l}^2 ext{log l})$ with $2 ext{l}$ resource augmentation

Abstract

In this paper, we study the weighted $k$-server problem on the uniform metric in both the offline and online settings. We start with the offline setting. In contrast to the (unweighted) $k$-server problem which has a polynomial-time solution using min-cost flows, there are strong computational lower bounds for the weighted $k$-server problem, even on the uniform metric. Specifically, we show that assuming the unique games conjecture, there are no polynomial-time algorithms with a sub-polynomial approximation factor, even if we use $c$-resource augmentation for $c < 2$. Furthermore, if we consider the natural LP relaxation of the problem, then obtaining a bounded integrality gap requires us to use at least $\ell$ resource augmentation, where $\ell$ is the number of distinct server weights. We complement these results by obtaining a constant-approximation algorithm via LP rounding, with a resource augmentation of $(2+\epsilon)\ell$ for any constant $\epsilon > 0$. In the online setting, an $\exp(k)$ lower bound is known for the competitive ratio of any randomized algorithm for the weighted $k$-server problem on the uniform metric. In contrast, we show that $2\ell$-resource augmentation can bring the competitive ratio down by an exponential factor to only $O(\ell^2 \log \ell)$. Our online algorithm uses the two-stage approach of first obtaining a fractional solution using the online primal-dual framework, and then rounding it online.