The k-Server with Preferences Problem

TL;DR

This paper introduces a generalized k-Server problem with preferences, analyzing the competitive ratios of algorithms in uniform and non-uniform metrics, revealing trade-offs and bounds for mixed request types.

Contribution

It models a new k-Server variant with request preferences, provides algorithms with competitive ratios, and establishes bounds and trade-offs for mixed request scenarios.

Findings

Competitive ratio of 3k-2 for one algorithm under certain request frequencies.

Lower bounds of 3k-2 and 2k-1 for the algorithms, respectively.

Adapted algorithms for non-uniform metrics with specific competitive ratios.

Abstract

The $k$-Server Problem covers plenty of resource allocation scenarios, and several variations have been studied extensively for decades. We present a model generalizing the $k$-Server Problem by preferences of the requests, where the servers are not identical and requests can express which specific servers should serve them. In our model, requests can either be answered by any server (general requests) or by a specific one (specific requests). If only general requests appear, the instance is one of the original $k$-Server Problem, and a lower bound for the competitive ratio of $k$ applies. If only specific requests appear, a solution with a competitive ratio of $1$ becomes trivial. We show that if both kinds of requests appear, the lower bound raises to $2k-1$. We study deterministic online algorithms and present two algorithms for uniform metrics. The first one has a competitive ratio dependent on the frequency of specific requests. It achieves a worst-case competitive ratio of $3k-2$ while it is optimal when only general requests appear or when specific requests dominate the input sequence. The second has a worst-case competitive ratio of $2k+14$. For the first algorithm, we show a lower bound of $3k-2$, while the second algorithm has a lower bound of $2k-1$ when only general requests appear. The two algorithms differ in only one behavioral rule that significantly influences the competitive ratio. We show that there is a trade-off between performing well against instances of the $k$-Server Problem and mixed instances based on the rule. Additionally, no deterministic online algorithm can be optimal for both kinds of instances simultaneously. Regarding non-uniform metrics, we present an adaption of the Double Coverage algorithm for $2$ servers on the line achieving a competitive ratio of $6$, and an adaption of the Work-Function-Algorithm achieving a competitive ratio of $4k$.