Before revision:  }[ . Specifically, our algorithm is deterministic and it achieves an approximation ratio of (1+\epsilon) f and an update time of O\left((f^2/\epsilon^3)+(f/\epsilon^2) \log C\right) . Our algorithm is based on the dynamic primal-dual framework, and it carefully combines the ideas from the dynamic vertex cover algorithm of Bhattacharya and Kulkarni [SODA ` 19] and the dynamic set cover algorithm of Bhattacharya et al.
After revision:  }[ . Specifically, our algorithm is deterministic and it achieves an approximation ratio of (1+\epsilon) f and an update time of 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] and the dynamic set cover algorithm of Bhattacharya et al.