The k-Server with Preferences Problem

The famous $k$-Server Problem covers plenty of resource allocation scenarios, and several variations have been studied extensively for decades. However, to the best of our knowledge, no research has considered the problem if the servers are not identical and requests can express which specific servers should serve them. Therefore, we present a new model generalizing the $k$-Server Problem by *preferences* of the requests and proceed to study it in a uniform metric space for deterministic online algorithms (the special case of paging). 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 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 since there is no freedom regarding the servers' movements. Perhaps counter-intuitively, 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 an adaptive 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 or only specific requests appear (competitive ratio of $k$ and $1$, respectively). The second has a fixed 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 algorithm has a lower bound of $2k-1$ when only general requests appear. The two 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. In other words, there is a trade-off between performing well against instances of the $k$-Server Problem and instances containing specific requests. We also show that no deterministic online algorithm can be optimal for both kinds of instances simultaneously.