Fairness in the k-Server Problem

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.