Dynamic Set Cover: Improved Amortized and Worst-Case Update Time
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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+ϵ)f-approximation algorithm with O(f log (Cn)/ϵ^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+ϵ)f-approximation ratio in O(flog^2 (Cn)/ϵ^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/poly(ϵ)) 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+ϵ)f-approximation ratio in O((f^2/ϵ^3)+(f/ϵ^2) log C) amortized update time: This result improves the previous O(f log (Cn)/ϵ^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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