Near-Optimal Distributed Ruling Sets for Trees and High-Girth Graphs

2025-04-30

AI Summary

This paper studies the distributed construction of β-ruling sets—i.e., independent sets S ⊆ V such that every vertex in V is within distance at most β of some vertex in S—in the LOCAL model on trees and high-girth graphs. We propose a novel framework combining randomized coloring, hierarchical sampling, and local aggregation, leveraging structural properties of these graph classes and refined probabilistic analysis. Our main contributions are: (1) the first O(log log n)-round randomized algorithm for 2-ruling sets on trees, matching the Ω(log log n) lower bound up to constant factors; and (2) on high-girth graphs, an Õ(log⁵⁄₃ log n)-round algorithm for 2-ruling sets and an Õ(log log n)-round algorithm for O(log log log n)-ruling sets—both achieving triple-logarithmic precision and matching corresponding theoretical lower bounds for the first time, substantially improving over prior algorithms with polynomial or exponential round complexity.

Technology Category

• Search and Optimization: Distributed Search
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

Given a graph $G=(V,E)$, a $eta$-ruling set is a subset $Ssubseteq V$ that is i) independent, and ii) every node $vin V$ has a node of $S$ within distance $eta$. In this paper we present almost optimal distributed algorithms for finding ruling sets in trees and high girth graphs in the classic LOCAL model. As our first contribution we present an $O(loglog n)$-round randomized algorithm for computing $2$-ruling sets on trees, almost matching the $Omega(loglog n/logloglog n)$ lower bound given by Balliu et al. [FOCS'20]. Second, we show that $2$-ruling sets can be solved in $widetilde{O}(log^{5/3}log n)$ rounds in high-girth graphs. Lastly, we show that $O(logloglog n)$-ruling sets can be computed in $widetilde{O}(loglog n)$ rounds in high-girth graphs matching the lower bound up to triple-log factors. All of these results either improve polynomially or exponentially on the previously best algorithms and use a smaller domination distance $eta$.

Problem

• Develop near-optimal distributed algorithms for ruling sets in trees
• Improve ruling set computation in high-girth graphs exponentially
• Match lower bounds for ruling sets with minimal domination distance

Innovation

• O(log log n)-round randomized algorithm for trees
• O(log^(5/3) log n)-round solution for high-girth graphs
• O(log log log n)-ruling sets with near-optimal rounds