Randomized $k$-server in polynomial time

TL;DR

This paper introduces a derandomization framework that transforms existing randomized $k$-server algorithms into polynomial-time algorithms with polylogarithmic competitive ratios on arbitrary metrics.

Contribution

It provides the first polynomial-time randomized $k$-server algorithm with a polylogarithmic competitive ratio, using a novel derandomization approach on hierarchically separated trees.

Findings

Achieves polynomial-time randomized $k$-server algorithm with polylogarithmic competitive ratio.

Introduces a derandomization framework reducing randomness to $O(\log k)$ bits.

Implications for advice complexity of the $k$-server problem.

Abstract

We study the design of computationally efficient randomized algorithms for the $k$-server problem. Existing randomized algorithms with the best known competitive ratios are, on the one hand, inherently implicit and, on the other hand, employ a rounding scheme that maintains a distribution over exponentially many configurations. In this work, we introduce a derandomization framework that transforms any randomized $k$-server algorithm on a hierarchically separated tree into one that uses only $O(\log k)$ random bits for request sequences of arbitrary length; hence maintaining a distribution over only polynomially many server configurations. Leveraging this black-box derandomization, we obtain the first polynomial-time randomized $k$-server algorithm on arbitrary $n$-point metrics with a polylogarithmic competitive ratio. Our results also have implications for the advice complexity of the $k$-server problem.