Fully Dynamic Set Cover via Hypergraph Maximal Matching: An Optimal Approximation Through a Local Approach

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 \emph{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 $O(f \cdot \log{n}/\epsilon^2)$ update time and $(1+\epsilon) \cdot f$ approximation, improving the $O(f² \cdot \log{n}/\epsilon^5)$ 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 [...]