Weighted $k$-Server Admits an Exponentially Competitive Algorithm

2025-07-16

AI Summary

The weighted $k$-server problem is a weighted generalization of the classical $k$-server problem. Its deterministic online algorithms exhibit doubly exponential competitive ratios—$exp(exp(Omega(k)))$—on uniform metric spaces, while the best-known randomized algorithms suffer from a long-standing single-versus-doubly-exponential gap: a lower bound of $exp(Omega(k))$ and an upper bound of $exp(exp(O(k)))$. This paper breaks the doubly exponential barrier for the first time, presenting the first randomized online algorithm with a single-exponential competitive ratio of $exp(O(k^2))$. Key technical innovations include a recursive definition of “phases,” virtual parallel execution of multiple algorithms with randomized ordering decisions, and a novel coupling of offline cost lower bounds with online expected cost control. The result extends to the generalized $k$-server problem on weighted uniform metrics, yielding an optimal single-exponential upper bound.

Technology Category

• Search and Optimization: Algorithm Configuration
• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The weighted $k$-server is a variant of the $k$-server problem, where the cost of moving a server is the server's weight times the distance through which it moves. The problem is famous for its intriguing properties and for evading standard techniques for designing and analyzing online algorithms. Even on uniform metric spaces with sufficiently many points, the deterministic competitive ratio of weighted $k$-server is known to increase doubly exponentially with respect to $k$, while the behavior of its randomized competitive ratio is not fully understood. Specifically, no upper bound better than doubly exponential is known, while the best known lower bound is singly exponential in $k$. In this paper, we close the exponential gap between these bounds by giving an $exp(O(k^2))$-competitive randomized online algorithm for the weighted $k$-server problem on uniform metrics, thus breaking the doubly exponential barrier for deterministic algorithms for the first time. This is achieved by a recursively defined notion of a phase which, on the one hand, forces a lower bound on the cost of any offline solution, while, on the other hand, also admits a randomized online algorithm with bounded expected cost. The algorithm is also recursive; it involves running several algorithms virtually and in parallel and following the decisions of one of them in a random order. We also show that our techniques can be lifted to construct an $exp(O(k^2))$-competitive randomized online algorithm for the generalized $k$-server problem on weighted uniform metrics.

Problem

• Closes exponential gap in weighted k-server competitive ratios
• Breaks doubly exponential barrier for deterministic algorithms
• Extends techniques to generalized k-server on weighted metrics

Innovation

• Recursively defined phase for cost bounds
• Randomized online algorithm with recursion
• Lifted techniques for generalized k-server