We give new upper and lower bounds for the dynamic set cover problem. First, we give a (1 + ε)f -approximation for fully dynamic set cover in O(f 2 log n/ε 5 ) (amortized) update time, for any ǫ > 0, where f is the maximum number of sets that an element belongs to. In the decremental setting, the update time can be improved to O(f 2 /ε 5 ), while still obtaining an (1 + ε)f -approximation. These are the first algorithms that obtain an approximation factor linear in f for dynamic set cover, thereby almost matching the best bounds known in the offline setting and improving upon the previous best approximation of O(f 2 ) in the dynamic setting.To complement our upper bounds, we also show that a linear dependence of the update time on f is necessary unless we can tolerate much worse approximation factors. Using the recent distributed PCP-framework, we show that any dynamic set cover algorithm that has an amortized update time of O(f 1−ε ) must have an approximation factor that is Ω(n δ ) for some constant δ > 0 under the Strong Exponential Time Hypothesis.

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