Nearly Optimal Dynamic Set Cover: Breaking the Quadratic-in-\(f\) Time Barrier

The dynamic set cover problem has been subject to extensive research since the pioneering works of [Bhattacharya et al, 2015] and [Gupta et al, 2017]. The input is a set system \((U, S)\) on a fixed collection \(S\) of sets and a dynamic universe of elements, where each element appears in a most \(f\) sets and the cost of each set lies in the range \([1/C, 1]\), and the goal is to efficiently maintain an approximately-minimum set cover under insertions and deletions of elements. Most previous work considers the low-frequency regime, namely \(f = O(\log n)\), and this line of work has culminated with a deterministic \((1+\epsilon)f\)-approximation algorithm with amortized update time \(O(\frac{f^2}{\epsilon^3} + \frac{f}{\epsilon^2}\log C)\) [Bhattacharya et al, 2021]. In the high-frequency regime of \(f = \Omega(\log n)\), an \(O(\log n)\)-approximation algorithm with amortized update time \(O(f\log n)\) was given by [Gupta et al, 2017]. Interestingly, at the intersection of the two regimes, i.e., \(f = \Theta(\log n)\), the state-of-the-art results coincide: approximation \(\Theta(f) = \Theta(\log n)\) with amortized update time \(O(f^2) = O(f \log n) = O(\log^2 n)\). Up to this date, no previous work achieved update time of \(o(f^2)\). In this paper we break the \(\Omega(f^2)\) update time barrier via the following results: (1) \((1+\epsilon)f\)-approximation can be maintained in \(O\left(\frac{f}{\epsilon^3}\log^*f + \frac{f}{\epsilon^3}\log C\right) = O_{\epsilon,C}(f \log^* f)\) expected amortized update time; our algorithm works against an adaptive adversary. (2) \((1+\epsilon)f\)-approximation can be maintained deterministically in \(O\left(\frac{1}{\epsilon}f\log f + \frac{f}{\epsilon^3} + \frac{f\log C}{\epsilon^2}\right) = O_{\epsilon,C}(f \log f)\) amortized update time.