Fully Dynamic Maximal Independent Set with Sublinear in n Update Time.

The first fully dynamic algorithm for maintaining a maximal independent set (MIS) with update time that is sublinear in the number of edges was presented recently by the authors of this paper [Assadi et al., STOC'18]. The algorithm is deterministic and its update time is O(m3/4), where m is the (dynamically changing) number of edges. Subsequently, Gupta and Khan and independently Du and Zhang [arXiv, April 2018] presented deterministic algorithms for dynamic MIS with update times of O(m2/3) and [MATH HERE], respectively. Du and Zhang also gave a randomized algorithm with update time [MATH HERE]. Moreover, they provided some partial (conditional) hardness results hinting that the update time of m1/2−ε, and in particular n1−ε for n-vertex dense graphs, is a natural barrier for this problem for any constant ε > 0, for deterministic and randomized algorithms that satisfy a certain natural property. In this paper, we break this natural barrier and present the first fully dynamic (randomized) algorithm for maintaining an MIS with update time that is always sublinear in the number of vertices, namely, an [MATH HERE] expected amortized update. We also show that a simpler variant of our algorithm can already achieve an Õ(m1/3) expected amortized update time, which results in an improved performance over our [MATH HERE] update time algorithm for sufficiently sparse graphs, and breaks the m1/2 barrier of Du and Zhang for all values of m.