Fairness in the k-Server Problem

2025-12-24

AI Summary

This paper initiates the formal study of fairness in the $k$-server problem, introducing the $(alpha,eta)$-fairness notion: minimizing total movement cost while balancing load across servers. Methodologically, it designs an offline deterministic redistribution algorithm and an online randomized reduction framework, leveraging metric space diameter analysis and probabilistic techniques against oblivious adversaries. Theoretically, it proves that fairness can be achieved without compromising optimal or competitive performance; moreover, it establishes that Double Coverage satisfies nontrivial fairness only on lines and trees with $k=2$, exposing inherent limitations on general trees. Experimentally, the offline algorithm achieves $(1+varepsilon, O(mathrm{diam} cdot log k / varepsilon))$-fairness with additive cost overhead $O(mathrm{diam} cdot k log k / varepsilon)$; the online algorithm attains fairness with high probability against oblivious adversaries while preserving competitiveness.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $(α,β)$-fairness, where, for parameters $αge 1$ and $βge 0$, 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.

Problem

• Defines fairness in k-server problem to equitably distribute movement costs among servers.
• Develops algorithms achieving fairness without competitiveness loss in offline and online settings.
• Investigates fairness feasibility against adaptive adversaries in online k-server scenarios.

Innovation

• Defining fairness with parameters alpha and beta
• Transforming optimal solutions into fair ones
• Achieving fairness without competitiveness loss