Improved Algorithms and Lower Bounds for Parametrized Metrical Service Systems

Abstract

We consider the parametrized setting of the classical metrical service system (MSS) problem first studied by Bubeck and Rabani (APPROX/RANDOM 2020). In this setting, the adversary is restricted to a set of $m$ distinct request types, known to the algorithm in advance. The goal is to obtain competitive ratio bounds in terms of $m$. In this work, we make significant progress in understanding the landscape of parametrized MSS and resolve several open problems from Bubeck and Rabani. Our first main result is a tight bound for parametrized MSS on weighted stars. Previously, Bubeck and Rabani gave a randomized lower bound of $\hat{I}\text{c◯}(m)$ and deterministic upper bound of $O(2^m)$. We show that, surprisingly, a deterministic $O(m)$-competitive algorithm exists, matching the randomized lower bound. Our key insight is an interval covering formulation of MSS on weighted stars which enables an application of the primal-dual method. Our second main contribution is an improved lower bound construction for parametrized MSS on hierarchically separated trees (HSTs). Bubeck and Rabani's construction gave a $\ddot{I}‰(1)$ lower bound when $m\ge 6$. Our improved lower bounds are tight for $2$-level HSTs and also rule out $O(1)$-competitive algorithms on HSTs when the parameter $m\ge 4$. We also complement these results by giving a deterministic $O(1)$-competitive algorithm on general metrics when $m=2$ while showing that it is impossible when $m\ge 3$.

To appear in APPROX 2026