Fully Dynamic Maximal Independent Set in Expected Poly-Log Update Time

In the fully dynamic maximal independent set (MIS) problem our goal is to maintain an MIS in a given graph G while edges are inserted and deleted from the graph. The first non-trivial algorithm for this problem was presented by Assadi, Onak, Schieber, and Solomon [STOC 2018] who obtained a deterministic fully dynamic MIS with O(m
^3/4
) update time. Later, this was independently improved by Du and Zhang and by Gupta and Khan [arXiv 2018] to Õ(m
^2/3
) update time
¹
Du and Zhang [arXiv 2018] also presented a randomized algorithm against an oblivious adversary with Õ(√m) update time. The current state of art is by Assadi, Onak, Schieber, and Solomon [SODA 2019] who obtained randomized algorithms against oblivious adversary with Õ(√n) and Õ(m
^1/3
) update times. In this paper, we propose a dynamic randomized algorithm against oblivious adversary with expected worst-case update time of O(log
⁴
n). As a direct corollary, one can apply the black-box reduction from a recent work by Bernstein, Forster, and Henzinger [SODA 2019] to achieve O(log
⁶
n) worst-case update time with high probability. This is the first dynamic MIS algorithm with very fast update time of poly-log.