The k-Server with Preferences Problem

The famous $k$-Server Problem covers plenty of resource allocation scenarios, and several variations have been studied extensively. However, to the best of our knowledge, no research has considered the problem if the servers are not identical and requests can express which servers should serve them. Therefore, we present a new model generalizing the $k$-Server Problem by preferences of the requests and study it in uniform metrics for deterministic online algorithms. In our model, requests can either demand to 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 $k$-Server Problem, and a lower bound for the competitive ratio of $k$ applies. If only specific requests appear, a competitive ratio of $1$ becomes trivial since there is no freedom regarding the servers' movements. We show that if both kinds of requests appear, the lower bound raises to $2k-1$. We study deterministic online algorithms in uniform metrics and present two algorithms. 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 (ratio of $k$) or specific requests dominate the input. The second has a close-to-optimal worst-case competitive ratio of $2k+14$. For the first algorithm, we show a lower bound of $3k-2$, while the second one has one of $2k-1$ when only general requests appear. Both algorithms differ in only one behavioral rule for each server that significantly influences the competitive ratio. Each server acting according to the rule allows approaching the worst-case lower bound, while it implies an increased lower bound for $k$-Server instances. Thus, there is a trade-off between performing well against instances of the $k$-Server Problem and ones containing specific requests.