# Symmetry Breaking in the Congest Model: Time- and Message-Efficient Algorithms for Ruling Sets

## TL;DR

This paper introduces faster algorithms for ruling sets in the CONGEST model, breaking traditional time and message complexity barriers for symmetry breaking problems like MIS, with new algorithms for 2- and 3-ruling sets.

## Contribution

The paper presents the first sub-logarithmic time algorithms for 2- and 3-ruling sets in the CONGEST model and introduces a nearly linear message complexity algorithm for 2-ruling sets.

## Key findings

- 3-ruling sets computed in $O(rac{ ext{log} n}{ ext{log} ext{log} n})$ rounds
- 2-ruling sets computed in $O( ext{log} ext{Delta} imes ( ext{log} n)^{1/2 + ext{epsilon}} + rac{ ext{log} n}{ ext{log} ext{log} n})$ rounds
- Message complexity for 2-ruling sets is $O(n ext{log}^2 n)$, nearly linear in $n$

## Abstract

We study local symmetry breaking problems in the CONGEST model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. A $\beta$-ruling set is an independent set such that every node in the graph is at most $\beta$ hops from a node in the independent set. Our work is motivated by the following central question: can we break the $\Theta(\log n)$ time complexity barrier and the $\Theta(m)$ message complexity barrier in the CONGEST model for MIS or closely-related symmetry breaking problems? We present the following results: - Time Complexity: We show that we can break the $O(\log n)$ "barrier" for 2- and 3-ruling sets. We compute 3-ruling sets in $O\left(\frac{\log n}{\log \log n}\right)$ rounds with high probability (whp). More generally we show that 2-ruling sets can be computed in $O\left(\log \Delta \cdot (\log n)^{1/2 + \varepsilon} + \frac{\log n}{\log\log n}\right)$ rounds for any $\varepsilon > 0$, which is $o(\log n)$ for a wide range of $\Delta$ values (e.g., $\Delta = 2^{(\log n)^{1/2-\varepsilon}}$). These are the first 2- and 3-ruling set algorithms to improve over the $O(\log n)$-round complexity of Luby's algorithm in the CONGEST model. - Message Complexity: We show an $\Omega(n^2)$ lower bound on the message complexity of computing an MIS (i.e., 1-ruling set) which holds also for randomized algorithms and present a contrast to this by showing a randomized algorithm for 2-ruling sets that, whp, uses only $O(n \log^2 n)$ messages and runs in $O(\Delta \log n)$ rounds. This is the first message-efficient algorithm known for ruling sets, which has message complexity nearly linear in $n$ (which is optimal up to a polylogarithmic factor).