Admission Control to Minimize Rejections and Online Set Cover with Repetitions (0803.2842v1)

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.

• The paper presents novel competitive algorithms that reduce rejections in admission control by leveraging preemptive online decision-making.
• It establishes an O(log²(mc)) bound for weighted cases and O(log m log c) for unweighted cases, enhancing network resource management.
• The work extends to online set cover with repetitions, delivering an O(log m log n) randomized algorithm that matches established lower bounds.

Admission Control to Minimize Rejections and Online Set Cover with Repetitions

The paper by Alon, Azar, and Gutner focuses on tackling the admission control problem within general networks, alongside addressing the online set cover problem with repetitions. These two interlinked challenges are explored through the development and analysis of competitive randomized algorithms designed to minimize rejections and optimize coverage.

Admission Control Problem

In network management, the admission control problem involves managing incoming communication requests by deciding acceptance or rejection, while respecting network capacities. Traditional analyses have focused on maximizing accepted requests. However, this paper shifts focus to minimizing rejections, a perspective relevant for networks where rejections should be infrequent events.

Key contributions include:

• Competitive Algorithms: The authors present an $O(\log^2 (mc))$-competitive randomized algorithm for the weighted case and an $O(\log m \log c)$-competitive randomized algorithm for the unweighted case, where $m$ represents the number of edges and $c$ the maximum edge capacity. These results settle previously open questions regarding competitive strategies for minimizing rejections.
• Preemptive Online Algorithms: Allowances for preemption enhance competitiveness, contrasted against straightforward lower bounds when preemption is not possible. The methodology relies on each request arriving with its path predefined, permitting strategic rejection decisions.

Online Set Cover with Repetitions

This problem extends the classical online set cover problem by allowing repeated requests for set elements. The authors explore the problem through a reduction to the admission control scenario:

• Algorithmic Bounds: They develop an $O(\log m \log n)$-competitive randomized algorithm for the online set cover with repetitions. Remarkably, this matches lower bounds established by prior literature, aligning with Feige and Korman’s findings under certain complexity assumptions.
• Deterministic Bicriteria Approach: For scenarios requiring deterministic solutions, an $O(\log m \log n)$-competitive algorithm is introduced that guarantees nearly complete coverage. This is particularly useful for ensuring robustness in algorithmic performance when handle coverage frequencies less than the optimal.

Implications and Future Direction

The results have significant practical and theoretical implications. For network operators, the ability to minimally reject requests enhances service consistency and user satisfaction, providing a nuanced approach to network resource allocation. Theoretically, these advances contribute to understanding competitive strategies in online algorithms and set a course for future exploration in superpolynomial contexts and potential further derandomization efforts.

The paper opens interesting avenues for future research, particularly in refining algorithmic performance where requests are not confined to simple paths or exploring if current randomized bounds translate effectively into deterministic regimes without efficiency trade-offs.