Fully Dynamic Maximal Independent Set with Sublinear in n Update Time

TL;DR

This paper introduces a randomized fully dynamic algorithm for maintaining a maximal independent set with expected amortized update time sublinear in the number of vertices, breaking previous barriers and improving performance for sparse graphs.

Contribution

It presents the first fully dynamic randomized MIS algorithm with update time sublinear in n, specifically (()), surpassing prior deterministic and randomized bounds.

Findings

Achieves (()) expected amortized update time

Simpler variant attains (()) update time for sparse graphs

Breaks the m^{1/2} barrier for all m values

Abstract

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 $O(m^{3/4})$, 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 $O(m^{2/3})$ and $O(m^{2/3} \sqrt{\log m})$, respectively. Du and Zhang also gave a randomized algorithm with update time $\widetilde{O}(\sqrt{m})$. Moreover, they provided some partial (conditional) hardness results hinting that update time of $m^{1/2-\epsilon}$, and in particular $n^{1-\epsilon}$ for $n$-vertex dense graphs, is a natural barrier for this problem for any constant $\epsilon >0$, 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 $\widetilde{O}(\sqrt{n})$ expected amortized update time algorithm. We also show that a simpler variant of our algorithm can already achieve an $\widetilde{O}(m^{1/3})$ expected amortized update time, which results in an improved performance over our $\widetilde{O}(\sqrt{n})$ update time algorithm for sufficiently sparse graphs, and breaks the $m^{1/2}$ barrier of Du and Zhang for all values of $m$.