Deterministic Distributed Ruling Sets of Line Graphs

TL;DR

This paper introduces efficient deterministic distributed algorithms for computing ruling sets in line graphs and related structures, achieving near-optimal round complexity in the CONGEST model.

Contribution

It presents the first simple deterministic algorithms for ruling sets of line graphs and extends these methods to graphs with bounded diversity and hypergraphs, improving efficiency.

Findings

Deterministic $(2,2)$-ruling edge set algorithm in $O(\log^* n)$ rounds.

Extension to graphs with bounded diversity with $O( ext{diversity} + \log^* n)$ rounds.

Fast ruling set algorithms for general graphs with various parameters, matching known bounds.

Abstract

An $(\alpha,\beta)$-ruling set of a graph $G=(V,E)$ is a set $R\subseteq V$ such that for any node $v\in V$ there is a node $u\in R$ in distance at most $\beta$ from $v$ and such that any two nodes in $R$ are at distance at least $\alpha$ from each other. The concept of ruling sets can naturally be extended to edges, i.e., a subset $F\subseteq E$ is an $(\alpha,\beta)$-ruling edge set of a graph $G=(V,E)$ if the corresponding nodes form an $(\alpha,\beta)$-ruling set in the line graph of $G$. This paper presents a simple deterministic, distributed algorithm, in the $\mathsf{CONGEST}$ 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 $(2,O(\mathcal{D}))$-ruling sets on graphs with diversity $\mathcal{D}$ in $O(\mathcal{D}+\log^* n)$ rounds. This also implies a fast, deterministic $(2,O(\ell))$-ruling edge set algorithm for hypergraphs with rank at most $\ell$. Furthermore, we provide a ruling set algorithm for general graphs that for any $B\geq 2$ computes an $\big(\alpha, \alpha \lceil \log_B n \rceil \big)$-ruling set in $O(\alpha \cdot B \cdot \log_B n)$ rounds in the $\mathsf{CONGEST}$ model. The algorithm can be modified to compute a $\big(2, \beta \big)$-ruling set in $O(\beta \Delta^{2/\beta} + \log^* n)$ rounds in the $\mathsf{CONGEST}$~ model, which matches the currently best known such algorithm in the more general $\mathsf{LOCAL}$ model.