Fully Dynamic Maximal Independent Set with Polylogarithmic Update Time

Abstract: We present the first algorithm for maintaining a maximal independent set (MIS) of a fully dynamic graph---which undergoes both edge insertions and deletions---in polylogarithmic time. Our algorithm is randomized and, per update, takes $$O(\log² \Delta \cdot \log² n)$$ expected time. Furthermore, the algorithm can be adjusted to have $$O(\log² \Delta \cdot \log⁴ n)$$ worst-case update-time with high probability. Here, $n$ denotes the number of vertices and $\Delta$ is the maximum degree in the graph. The MIS problem in fully dynamic graphs has attracted significant attention after a breakthrough result of Assadi, Onak, Schieber, and Solomon [STOC'18] who presented an algorithm with $O(m<sup>{3/4})$ update-time (and thus broke the natural $\Omega(m)$ barrier) where $m$ denotes the number of edges in the graph. This result was improved in a series of subsequent papers, though, the update-time remained polynomial. In particular, the fastest algorithm prior to our work had $\widetilde{O}(\min{\sqrt{n}, m<sup>{1/3}})$ update-time [Assadi et al. SODA'19]. Our algorithm maintains the lexicographically first MIS over a random order of the vertices. As a result, the same algorithm also maintains a 3-approximation of correlation clustering. We also show that a simpler variant of our algorithm can be used to maintain a random-order lexicographically first maximal matching in the same update-time.