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 O(m2/3 √log m), respectively. Du and Zhang also gave a randomized algorithm with update time widetildeO(√m). Moreover, they provided some partial (conditional) hardness results hinting that 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 both 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 widetildeO(√n) expected amortized update time algorithm. We also show that a simpler variant of our algorithm can already achieve an widetildeO(m1/3) expected amortized update time, which results in an improved performance over our widetildeO(√n) update time algorithm for sufficiently sparse graphs, and breaks the m1/2 barrier of Du and Zhang for all values of m.