Admission Control to Minimize Rejections and Online Set Cover with Repetitions

TL;DR

This paper introduces online algorithms for admission control in networks that aim to minimize rejections, providing competitive solutions for weighted and unweighted cases, and addresses an open problem in the field.

Contribution

It presents the first competitive randomized algorithms for preemptive admission control minimizing rejections, solving an open problem in network request management.

Findings

Achieves $O(\log^2 (mc))$-competitiveness for weighted networks.

Provides $O(\log m \log c)$-competitive algorithm for unweighted networks.

Addresses the open problem posed by Blum, Kalai, and Kleinberg in 2001.

Abstract

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