Fully Dynamic Set Cover via Hypergraph Maximal Matching: An Optimal Approximation Through a Local Approach
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In the (fully) dynamic set cover problem, we have a collection of m sets from a universe of size n that undergo element insertions and deletions; the goal is to maintain an approximate set cover of the universe after each update. We give an O(f^2) update time algorithm for this problem that achieves an f-approximation, where f is the maximum number of sets that an element belongs to; under the unique games conjecture, this approximation is best possible for any fixed f. This is the first algorithm for dynamic set cover with approximation ratio that exactly matches f (as opposed to almost f in prior work), as well as the first one with runtime independent of n,m (for any approximation factor of o(f^3)). Prior to our work, the state-of-the-art algorithms for this problem were O(f^2) update time algorithms of Gupta et al. [STOC'17] and Bhattacharya et al. [IPCO'17] with O(f^3) approximation, and the recent algorithm of Bhattacharya et al. [FOCS'19] with O(f ·logn/ϵ^2) update time and (1+ϵ) · f approximation, improving the O(f^2 ·logn/ϵ^5) bound of Abboud et al. [STOC'19]. The key technical ingredient of our work is an algorithm for maintaining a maximal matching in a dynamic hypergraph of rank r, where each hyperedge has at most r vertices, which undergoes hyperedge insertions and deletions in O(r^2) amortized update time; our algorithm is randomized, and the bound on the update time holds in expectation and with high probability. This result generalizes the maximal matching algorithm of Solomon [FOCS'16] with constant update time in ordinary graphs to hypergraphs, and is of independent merit; the previous state-of-the-art algorithms for set cover do not translate to (integral) matchings for hypergraphs, let alone a maximal one. Our quantitative result for the set cover problem is [...]
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