Polynomial Lower Bound for Distributed Graph Coloring in a Weak LOCAL Model

摘要

We show an Ω(Δ~(1/3-η/3)) lower bound on the runtime of any deterministic distributed (Ο)(Δ~(1+η))-graph coloring algorithm in a weak variant of the LOCAL model. In particular, given a network graph G = (V, E), in the weak LOCAL model nodes communicate in synchronous rounds and they can use unbounded local computation. The nodes have no identifiers, but instead, the computation starts with an initial valid vertex coloring. A node can broadcast a single message of unbounded size to its neighbors and receives the set of messages sent to it by its neighbors. The proof uses neighborhood graphs and improves their understanding in general such that it might help towards finding a lower (runtime) bound for distributed graph coloring in the standard LOCAL model.