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

TL;DR

It is proved that if there exists an $\alpha_1$-competitive algorithm for one version and there exists an $\alpha_2$-competitive algorithm for the other version, then there exists an $(\alpha_1\alpha_2)$-competitive algorithm for weighted $k-server on uniform metric spaces.

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 $α_1$-competitive algorithm for one version (which we call Weighted $k$-Server - Service Pattern Construction (W$k$S-SPC) and there exists an $α_2$-competitive algorithm for the other version (which we call Weighted $k$-server - Revealed Service Pattern (W$k$S-RSP)), then there exists an $(α_1α_2)$-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 $Ω(2^k)$ lower bound for weighted $k$-server, in fact, holds for W$k$S-RSP.

Figures (1)

• Figure 1: An illustration of \ref{['dichotomy-theorem-extn']} for $k=5$ and $\ell=3$. (a) Depicts $\mathcal{I}\xspace_t$ with a labeling $\gamma$ (colored in red) feasible with respect to $\rho_t$ such that $\gamma(L^5_t)=1$ and $\gamma(L^4_t)=2$. (b) Depicts the $3$-level service pattern $\mathcal{J}\xspace$ labeled with the restriction of $\gamma$, along with $\rho"$, the subsequence of $\rho'$ formed by removing all the requests to points $1$ and $2$. (d) Depicts the labeling $\gamma'$ of $\mathcal{J}\xspace$ feasible with respect to $\rho"$. (c) Shows the new labeling $\gamma_{\text{new}}$ constructed by overwriting $\gamma'$ onto $\gamma$ for every interval in $\mathcal{J}\xspace$.

Theorems & Definitions (37)

• Theorem 1: Composition Theorem
• Theorem 2
• Theorem 3
• Definition 1: Service Pattern and Levels BansalEK_FOCS17
• Definition 2: Labeling and Feasibility BansalEK_FOCS17
• Definition 3: Hierarchical Service Pattern BansalEK_FOCS17
• Definition 4: $\ell$-extension
• Definition 5: Weighted $k$-Server -- Service Pattern Construction (W$k$S-SPC)
• Definition 6: Weighted $k$-server -- Revealed Service Pattern (W$k$S-RSP)
• Theorem 3: Composition Theorem