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 $(α,β)$-fairness, where, for parameters $α\ge 1$ and $β\ge 0$, no server incurs more than an $α/k$-fraction of the total cost plus an additive term $β$. 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 $(α,β)$-fair solution for $α= 1 + \varepsilon$ and $β= O(\mathrm{diam} \cdot \log k / \varepsilon)$, 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.

翻译：我们首次对k-服务器问题中的公平性展开形式化研究，其目标不仅在于最小化总体移动成本，还要求在各服务器间公平分配成本。我们首先定义了广义的$(α,β)$-公平性概念：对于参数$α\ge 1$和$β\ge 0$，任何服务器承担的成本不超过总成本的$α/k$比例加上附加项$β$。随后我们证明，在离线与在线场景中实现公平性均不会损失竞争比。在离线场景中，我们提出确定性算法：对于任意$\varepsilon > 0$，可将任意最优解转化为$(α,β)$-公平解，其中$α= 1 + \varepsilon$，$β= O(\mathrm{diam} \cdot \log k / \varepsilon)$，且仅使解的成本增加$O(\mathrm{diam} \cdot k \log k / \varepsilon)$的附加项。此处$\mathrm{diam}$表示底层度量空间的直径。在线场景中我们给出类似结论：任何竞争算法均可转化为随机在线算法，该算法能以高概率对抗遗忘型对手实现公平性，同时保持竞争性且仅有微小损失。上述研究遗留关键问题：在线场景中能否通过确定性算法或随机算法，对抗完全自适应对手实现公平性？我们在此问题上取得进展：证明经典确定性双覆盖算法（DCA）在线度量空间及$k=2$时的树度量空间上具有公平性。但我们也给出否定结论：在一般树度量空间中，DCA无法在任何非平凡参数下保持公平性。