Fully Dynamic Maximal Independent Set with Sublinear Update Time

TL;DR

This paper introduces a deterministic algorithm for maintaining a maximal independent set in fully dynamic graphs with sublinear update time, improving efficiency and providing a distributed implementation.

Contribution

It presents the first deterministic, fully dynamic MIS algorithm with sublinear amortized update time and a distributed version with low message and round complexity.

Findings

Amortized update time is $O( ext{min}\{ ext{Δ}, m^{3/4} ight"])

Distributed implementation achieves $O( ext{min}\{ ext{Δ}, m^{3/4} ight"]) message complexity

Algorithm works under a fixed maximum degree bound and dynamic edge count

Abstract

A maximal independent set (MIS) can be maintained in an evolving $m$-edge graph by simply recomputing it from scratch in $O(m)$ time after each update. But can it be maintained in time sublinear in $m$ in fully dynamic graphs? We answer this fundamental open question in the affirmative. We present a deterministic algorithm with amortized update time $O(\min\{\Delta,m^{3/4}\})$, where $\Delta$ is a fixed bound on the maximum degree in the graph and $m$ is the (dynamically changing) number of edges. We further present a distributed implementation of our algorithm with $O(\min\{\Delta,m^{3/4}\})$ amortized message complexity, and $O(1)$ amortized round complexity and adjustment complexity (the number of vertices that change their output after each update). This strengthens a similar result by Censor-Hillel, Haramaty, and Karnin (PODC'16) that required an assumption of a non-adaptive oblivious adversary.