Online and dynamic algorithms for set cover

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.We give new results in both settings (all recourse and update time bounds are amortized):• In the update time setting, we obtain O(log n)-competitiveness with O(f log n) 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. The second result improves on previous work by removing an O(log n) factor in the update time bound. This has an important consequence: we obtain the first deterministic constant-competitive, constant update time algorithm for fully-dynamic vertex cover.• In the recourse setting, we show a competitive ratio of O(min{log n, f }) with constant recourse. The most relevant previous result is the O(log m log n) bound for online set cover in the insertion-only model with no recourse. Note that we can match the best offline bounds with O(1) recourse, something that is impossible in the classical online model.These results also yield, as corollaries, new results for the maximum k-coverage problem and the non-metric facility location problem in the fully dynamic 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.

A New Deterministic Algorithm for Dynamic Set Cover

“…Weighted? [19] O(log n) O(f log n) yes yes [19,9] O(f 3 ) O(f 2 ) yes yes [11] O(f 2 ) O(f log(m + n)) yes yes [1] (1 + ǫ)f O(f 2 log n/ǫ) no no Our result…”

“…Thus, we simultaneously (a) improve upon the update time of Abboud et al [1], (b) derandomize their result, and (c) extend their result to the weighted case. Our algorithm, together with the one in Gupta et al [19], settles an important open question in a line of work on dynamic set cover [9,11,19,1]. We can now get an O(min(log n, f ))-approximation using a deterministic algorithm with O(f log(Cn)) update time.…”

“…(1 + ǫ)f O(f log(Cn)/ǫ 2 ) yes yes Table 1: Summary of results on dynamic set cover of elements. 3 The O(f ) approximation ratio was recently achieved by Abboud et al [1] (improving upon the approximation factors of O(f 2 ) and higher by [11,19,9]). Abboud et al show how to maintain an (1 + ǫ)f -approximation in O(f 2 log n/ǫ) amortized update time.…”

“…[17,16,15,34,26]), it is natural to ask if one can also obtain these same guarantees in the dynamic setting with small update time. The O(log n) approximation ratio was already achieved in 2017 via greedy-like techniques by Gupta et al [19]. Their algorithm is deterministic and has O(f log n) update time.…”

Online Bipartite Matching with Amortized Replacements

“…There are many papers on online scheduling that try to minimize the number of job reassignments [1,11,24,25,27,28]. The model has also been studied in the context of flows [16,28], and there is a very recent result on online set cover with recourse [14].…”