A Decomposition Approach to the Weighted k-Server Problem

Abstract

A natural variant of the classical online k-server problem is the weighted k-server problem, where the cost of moving a server is its weight times the distance through which it moves. Despite its apparent simplicity, the weighted k-server problem is extremely poorly understood. Specifically, even on uniform metric spaces, finding the optimum competitive ratio of randomized algorithms remains an open problem - the best upper bound known is 2^{2^{k+O(1)}} due to a deterministic algorithm (Bansal et al., 2018), and the best lower bound known is Ω(2^k) (Ayyadevara and Chiplunkar, 2021).
With the aim of closing this exponential gap between the upper and lower bounds, we propose a decomposition approach for designing a randomized algorithm for weighted k-server on uniform metrics. Our first contribution includes two relaxed versions of the problem and a technique to obtain an algorithm for weighted k-server from algorithms for the two relaxed versions. Specifically, we prove that if there exists an α₁-competitive algorithm for one version (which we call Weighted k-Server - Service Pattern Construction) and there exists an α₂-competitive algorithm for the other version (which we call Weighted k-server - Revealed Service Pattern), then there exists an (α₁α₂)-competitive algorithm for weighted k-server on uniform metric spaces. Our second contribution is a 2^O(k²)-competitive randomized algorithm for Weighted k-server - Revealed Service Pattern. As a consequence, the task of designing a 2^poly(k)-competitive randomized algorithm for weighted k-server on uniform metrics reduces to designing a 2^poly(k)-competitive randomized algorithm for Weighted k-Server - Service Pattern Construction. Finally, we also prove that the Ω(2^k) lower bound for weighted k-server, in fact, holds for Weighted k-server - Revealed Service Pattern.