The online kk-server problem with rejection

In this work we investigate the online kk-server problem where each request has a penalty and it is allowed to reject the requests. The goal is to minimize the sum of the total distance moved by the servers and the total penalty of the rejected requests. We extend the work function algorithm to this more general model and prove that it is (4k−1)(4k−1)-competitive. We also consider the problem for special cases: we prove that the work function algorithm is 5-competitive if k=2k=2 and (2k+1)(2k+1)-competitive for any k≥1k≥1 if the metric space is the line. In the case of the line we also present the extension of the double-coverage algorithm and prove that it is 3k3k-competitive. This algorithm has worse competitive ratio than the work function algorithm but it is much faster and memoryless. Moreover we prove that for any metric space containing at least k+1k+1 points no online algorithm can have smaller competitive ratio than 2k+12k+1, and this shows that the work function algorithm has the smallest possible competitive ratio in the case of lines and also in the case k=2k=2.