Online and Dynamic Algorithms for Set Cover

TL;DR

This paper develops new algorithms for the fully dynamic set cover problem, achieving competitive ratios and update times that improve upon previous methods and match offline bounds with minimal recourse.

Contribution

It introduces novel dynamic algorithms with competitive ratios and update times, including the first to achieve a ratio independent of frequency f in this setting.

Findings

Achieves O(log n)-competitiveness with O(f log n) amortized update time.

Provides O(f^3)-competitiveness with O(f^2) update time.

Matches offline bounds with constant recourse in the recourse setting.

Abstract

In this paper, we study the set cover problem in the fully dynamic model. In this model, the set of active elements, i.e., those that must be covered at any given time, can change due to element arrivals and departures. The goal is to maintain an algorithmic solution that is competitive with respect to the current optimal solution. This model is popular in both the dynamic algorithms and online algorithms communities. The difference is in the restriction placed on the algorithm: in dynamic algorithms, the running time of the algorithm making updates (called update time) is bounded, while in online algorithms, the number of updates made to the solution (called recourse) is limited. In this paper we show the following results: In the update time setting, we obtain O(log n)-competitiveness with O(f log n) amortized update time, and O(f^3)-competitiveness with O(f^2) update time. The O(log n)-competitive algorithm is the first one to achieve a competitive ratio independent of f in this setting. In the recourse setting, we show a competitive ratio of O(min{log n,f}) with constant amortized recourse. Note that this matches the best offline bounds with just constant recourse, something that is impossible in the classical online model. Our results are based on two algorithmic frameworks in the fully-dynamic model that are inspired by the classic greedy and primal-dual algorithms for offline set cover. We show that both frameworks can be used for obtaining both recourse and update time bounds, thereby demonstrating algorithmic techniques common to these strands of research.