Deterministic distributed ruling sets of line graphs

Abstract

An (α, β)-ruling set of a graph G = (V, E) is a set R ⊆ V such that for any node v ∈ V there is a node u ∈ R in distance at most β from v and such that any two nodes in R are at distance at least α from each other. The concept of ruling sets can naturally be extended to edges, i.e., a subset F ⊆ E is an (α, β)-ruling edge set of a graph G = (V, E) if the corresponding nodes form an (α, β)-ruling set in the line graph of G. This paper presents a simple deterministic, distributed algorithm, in the CONGEST model, for computing (2, 2)-ruling edge sets in O(log ∗ n) rounds. Furthermore, we extend the algorithm to compute ruling sets of graphs with bounded diversity. Roughly speaking, the diversity of a graph is the maximum number of maximal cliques a vertex belongs to. We devise (2, O(D))-ruling sets on graphs with diversity D in O(D +log ∗ n) rounds. This also implies a fast, deterministic (2, O(ℓ))-ruling edge set algorithm for hypergraphs with rank at most ℓ. Furthermore, we provide a(ruling set algorithm) for general graphs that for any B ≥ 2 computes an α, α⌈log B n⌉-ruling set in O(α · B · log B n) rounds in the CONGEST model. The algorithm can be modified to compute a (2, β)-ruling set in O(βΔ 2/β + log ∗ n) rounds in the CONGEST model, which matches the currently best known such algorithm in the more general LOCAL model.