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We study the admission control problem in general networks. Communicationrequests arrive over time, and the online algorithm accepts or rejects eachrequest while maintaining the capacity limitations of the network. Theadmission control problem has been usually analyzed as a benefit problem, wherethe goal is to devise an online algorithm that accepts the maximum number ofrequests possible. The problem with this objective function is that evenalgorithms with optimal competitive ratios may reject almost all of therequests, when it would have been possible to reject only a few. This could beinappropriate for settings in which rejections are intended to be rare events.  In this paper, we consider preemptive online algorithms whose goal is tominimize the number of rejected requests. Each request arrives together withthe path it should be routed on. We show an $O(\log^2 (mc))$-competitiverandomized algorithm for the weighted case, where $m$ is the number of edges inthe graph and $c$ is the maximum edge capacity. For the unweighted case, wegive an $O(\log m \log c)$-competitive randomized algorithm. This settles anopen question of Blum, Kalai and Kleinberg raised in \cite{BlKaKl01}. We notethat allowing preemption and handling requests with given paths are essentialfor avoiding trivial lower bounds.

Admission Control to Minimize Rejections and Online Set Cover with Repetitions