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 $f=\Omega(\log n)$, 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 $(1+\epsilon)f$-approximation algorithm with $O(f \log (Cn)/\epsilon^2)$ 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. $(1+\epsilon)f$-approximation ratio in $O(f\log^2 (Cn)/\epsilon^3)$ 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 $O(\log^3 n/\text{poly}(\epsilon))$ worst-case update time for unweighted dynamic vertex cover (i.e., when $f=2$ and $C=1$) by Bhattacharya et al. [SODA'17]. 2. $(1+\epsilon)f$-approximation ratio in $O\left((f^2/\epsilon^3)+(f/\epsilon^2) \log C\right)$ amortized update time: This result improves the previous $O(f \log (Cn)/\epsilon^2)$ 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 $C = 1$).