We consider the minimum set cover problem in a dynamic setting. Here, we are given a universe of elements \mathcal{U and a collection of sets S \subseteq 2^{U} as input, where each set s \in S has a cost 1/C < c_s < 1 for some parameter C > 1. The input keeps changing via a sequence of updates. Each update inserts/deletes an element in the universe, and specifies the sets in \S containing that element. The goal is to maintain a set cover of (approximately) minimum cost with as small update time as possible. Let ndenote the }  number of sets in \mathcal{S number of elements in the universe and the maximum frequency of an element across all the updates.This problem has received significant attention in recent years in the dynamic algorithms community. Gupta et al. STOC `17%DIFDELCMD < ] %%% designed a deterministic \Theta(\log m )-approximation algorithm for this problem that has an update time of O(f \log m) . On the other hand ] , the line of work by Bhattacharya et al. ICALP `15%DIFDELCMD < ]%%% , Bhattacharya et al. IPCO `17%DIFDELCMD < ]%%% , Gupta et al. [STOC ` 17], Abboud et al. [STOC ` 19]  and Bhattacharya et al. [ FOCS ` 19] very recently led to a deterministic (1+\epsilon)f-approximation algorithm with O(f \log ( Cm )/\epsilon^2)  update time. In this paper , we obtain the first dynamic algorithm for this problem with near-optimal approximation ratio whose update timeis independent of m, n(\epsilon)) worst-case update time for unweighted dynamic vertex cover (i.e., when f=2 and C=1) by Bhattacharya et al. }[ . 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. FOCS `19 .
In the dynamic minimum set cover problem and a collection of sets S \subseteq 2^{U} as input, where each set s \in S has a cost 1/C < c_s < 1 for some parameter C > 1. The input keeps changing via a sequence of updates. Each update inserts/deletes an element in the universe, and specifies the sets in \S containing that element. The goal is to maintain a set cover of (approximately) minimum cost with as small update time as possible. Let ndenote the } , 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 %DIFDELCMD < ] %%% , 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 %DIFDELCMD < ]%%% %DIFDELCMD < ]%%% 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) .
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