Fairness in the k-Server Problem

TL;DR

This paper introduces a formal notion of fairness in the k-server problem, demonstrating that fairness can be achieved without sacrificing competitiveness in both offline and online settings, with some limitations on certain metrics.

Contribution

It defines a new fairness framework for the k-server problem and provides algorithms that achieve fairness without losing competitiveness, along with negative results on certain metrics.

Findings

Fairness can be integrated without loss of competitiveness in offline algorithms.

Online algorithms can be transformed to be fair with high probability while remaining competitive.

The classic DCA algorithm is fair on line and 2-server tree metrics but not on general tree metrics.

Abstract

We initiate a formal study of fairness for the $k$-server problem, where the objective is not only to minimize the total movement cost, but also to distribute the cost equitably among servers. We first define a general notion of $(\alpha,\beta)$-fairness, where, for parameters $\alpha \ge 1$ and $\beta \ge 0$, no server incurs more than an $\alpha/k$-fraction of the total cost plus an additive term $\beta$. We then show that fairness can be achieved without a loss in competitiveness in both the offline and online settings. In the offline setting, we give a deterministic algorithm that, for any $\varepsilon > 0$, transforms any optimal solution into an $(\alpha,\beta)$-fair solution for $\alpha = 1 + \varepsilon$ and $\beta = O(\mathrm{diam} \cdot \log k / \varepsilon)$, while increasing the cost of the solution by just an additive $O(\mathrm{diam} \cdot k \log k / \varepsilon)$ term. Here $\mathrm{diam}$ is the diameter of the underlying metric space. We give a similar result in the online setting, showing that any competitive algorithm can be transformed into a randomized online algorithm that is fair with high probability against an oblivious adversary and still competitive up to a small loss. The above results leave open a significant question: can fairness be achieved in the online setting, either with a deterministic algorithm or a randomized algorithm, against a fully adaptive adversary? We make progress towards answering this question, showing that the classic deterministic Double Coverage Algorithm (DCA) is fair on line metrics and on tree metrics when $k = 2$. However, we also show a negative result: DCA fails to be fair for any non-vacuous parameters on general tree metrics.