A Decomposition Approach to the Weighted $k$-server Problem

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 $\Omega(2^k)$ (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 $\alpha_1$-competitive algorithm for one version (which we call Weighted $k$-Server - Service Pattern Construction (W$k$S-SPC) and there exists an $\alpha_2$-competitive algorithm for the other version (which we call Weighted $k$-server - Revealed Service Pattern (W$k$S-RSP)), then there exists an $(\alpha_1\alpha_2)$-competitive algorithm for weighted $k$-server on uniform metric spaces. Our second contribution is a $2^{O(k^2)}$-competitive randomized algorithm for W$k$S-RSP. 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 W$k$S-SPC. Finally, we also prove that the $\Omega(2^k)$ lower bound for weighted $k$-server, in fact, holds for W$k$S-RSP.