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 $O(f^2 \cdot \log{n}/\epsilon^5)$ 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 [...]
翻译:在 { 完全 动态设置 覆盖问题 { 动态设置中, 我们从规模的宇宙中收集 $1 美元, 以16 美元 进行元素插入和删除; 目标是在每次更新后保持一个大致的宇宙覆盖。 我们给这个问题一个$( f) $ 更新时间算法, 以达到 $( 3) 的匹配值。 在我们工作之前, 任何固定( f) 的游戏预测下, 任何固定( f) 美元 都可能使用这种近似 。 这是第一个以 16 美元 元素插入和删除 ; 目标是在每次更新后保持 $( f) 美元 ; 我们的正常=2 美元 更新时间算法 ; 以 数( f) 美元 更新 Gupta 和 Al. [ ST' 美元 和 B. 美元 最新更新的 美元 。