Admission Control to Minimize Rejections

Admission control call control is a well studied online prob lem We are given a xed graph with edge capacities and must process a sequence of calls that arrive over time accepting some and rejecting others in order to stay within capacity limitations of the network In the standard theoretical formulation this problem is analyzed as a bene t problem the goal is to devise an online algorithm that accepts at least a reasonable fraction of the maximum number of calls that could possi bly have been accepted in hindsight This formulation however has the property that even algorithms with optimal competitive ratios typically O log n where n is the number of nodes may end up rejecting the vast majority of calls even when it would have been possible in hindsight to reject only very few In this paper we instead consider the goal of approximately minimizing the number of calls rejected This is much more appropriate for settings in which rejections are intended to be rare events In order to avoid trivial lower bounds we assume preemption is allowed and that calls are given to the algorithm as xed paths We show that in a number of cases we can in fact achieve a competitive ratio of for rejections so if the optimal in hindsight rejects then we reject if the optimal rejects r then we reject at most r For other cases we get worse but nontrivial bounds For the most general case of xed paths in arbitrary graphs with arbitrary edge capacities we achieve matching p m upper and lower bounds where m is the number of edges We also show a connection between these problems and online versions of the vertex cover and set cover problems our factor results give approximations to slight generalizations of the vertex cover problem much as AAA show hardness results for the bene t version based on the hardness of approximability of independent set