Dynamic set cover: Improved amortized and worst-case update time

In the dynamic minimum set cover problem, a challenge is to minimize the update time while guaranteeing close to the optimal min(O(log n), f) approximation factor. (Throughout, m, n, f, and C are parameters denoting the maximum number of sets, number of elements, frequency, and the cost range.) In the high-frequency range, when f = Ω(log n), this was achieved by a deterministic O(log n)-approximation algorithm with O(f log n) amortized update time [Gupta et al. STOC'17]. In the low-frequency range, the line of work by Gupta et al. [STOC'17], Abboud et al. [STOC'19], and Bhattacharya et al. [ICALP'15, IPCO'17, FOCS'19] led to a deterministic (1 + ∊) f-approximation algorithm with O(f log(Cn)/∊2) amortized update time. In this paper we improve the latter update time and provide the first bounds that subsume (and sometimes improve) the state-of-the-art dynamic vertex cover algorithms. We obtain: (1) (1 + ∊) f-approximation ratio in O(f log2(Cn)/∊3) worst-case update time: No non-trivial worst-case update time was previously known for dynamic set cover. Our bound subsumes and improves by a logarithmic factor the O(log3 n/poly(∊)) worst-case update time for unweighted dynamic vertex cover (i.e., when f = 2 and C = 1) by Bhattacharya et al. [SODA'17]. (2) (1 + ∊) f-approximation ratio in O ((f2/∊3) + (f/∊2) log C) amortized update time: This result improves the previous O(f log (Cn)/∊2) update time bound for most values of f in the low-frequency range, i.e. whenever f = o(log n). It is the first that is independent of m and n. It subsumes the constant amortized update time of Bhattacharya and Kulkarni [SODA'19] for unweighted dynamic vertex cover (i.e., when f = 2 and C = 1). These results are achieved by leveraging the approximate complementary slackness and background schedulers techniques. These techniques were used in the local update scheme for dynamic vertex cover. Our main technical contribution is to adapt these techniques within the global update scheme of Bhattacharya et al. [FOCS'19] for the dynamic set cover problem.