Admission Control to Minimize Rejections

Abstract

Admission control (call control) is a well-studied online problem. We are given a fixed 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 benefit problem: The goal is to devise an online algorithm that accepts at least a reasonable fraction of the maximum number of calls that could possibly 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 fixed paths. We show that in a number of cases, we can in fact achieve a competitive ratio of 2 for rejections (so if the optimal in hindsight rejects 0, then we reject 0; if the optimal rejects r, then we reject at most 2r). For other cases, we get worse, but nontrivial, bounds. For the most general case of fixed paths in arbitrary graphs with arbitrary edge capacities, we achieve matching $\Theta(\sqrt{m})$ 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-2 results give 2-approximations to slight generalizations of the vertex cover problem, much as [Alon et al. 99] show hardness results for the benefit version based on the hardness of approximability of independent set).