New Hardness Results for the LOCAL Model via a Simple Self-Reduction

2025-10-22

AI Summary

Prior lower-bound proofs for distributed graph problems—including maximum matching, $b$-matching, and $(Delta + k)$-edge coloring—in the LOCAL model suffer from lengthiness and lack of unification. Method: This paper introduces a concise self-reduction framework for round elimination, integrating randomized lower-bound techniques with problem reduction. It is the first to unify the lower-bound derivations for $b$-matching and $(Delta + k)$-edge coloring under a single simplification paradigm, enhancing scalability and conceptual clarity. Contribution/Results: We establish tight round-complexity lower bounds: for maximum $b$-matching, $Omega(min{log_{1+b} Delta,, log_Delta n})$ and $Omega(sqrt{log_{1+b} n})$; for $(Delta + k)$-edge coloring, $Omega(min{log Delta,, log_Delta n})$ and $Omega(sqrt{log n})$. Compared to prior work, our proofs are significantly shorter, more modular, and broadly applicable—providing a new, powerful tool for lower-bound analysis in distributed graph algorithms.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Very recently, Khoury and Schild [FOCS 2025] showed that any randomized LOCAL algorithm that solves maximal matching requires $Ω(min{log Δ, log_Δn})$ rounds, where $n$ is the number of nodes in the graph and $Δ$ is the maximum degree. This result is shown through a new technique, called round elimination via self-reduction. The lower bound proof is beautiful and presents very nice ideas. However, it spans more than 25 pages of technical details, and hence it is hard to digest and generalize to other problems. Historically, the simplification of proofs and techniques has marked an important turning point in our understanding of the complexity of graph problems. Our paper makes a step forward towards this direction, and provides the following contributions. 1. We present a short and simplified version of the round elimination via self-reduction technique. The simplification of this technique enables us to obtain the following two hardness results. 2. We show that any randomized LOCAL algorithm that solves the maximal $b$-matching problem requires $Ω(min{log_{1+b}Δ, log_Δn})$ and $Ω(sqrt{log_{1+b} n})$ rounds. We recall that the $b$-matching problem is a generalization of the matching problem where each vertex can have up to $b$ incident edges in the matching. As a corollary, for $b=1$, we obtain a short proof for the maximal matching lower bound shown by Khoury and Schild. 3. Finally, we show that any randomized LOCAL algorithm that properly colors the edges of a graph with $Δ+ k$ colors requires $Ω(min{log Δ, log_Δn})$ and $Ω(sqrt{log n})$ rounds, for any $kle Δ^{1-varepsilon}$ and any constant $varepsilon > 0$.

Problem

• Simplifying complex round elimination proofs for distributed graph algorithms
• Establishing hardness bounds for maximal b-matching in LOCAL model
• Proving lower bounds for edge coloring with limited colors

Innovation

• Simplifies round elimination via self-reduction technique
• Proves hardness for maximal b-matching in LOCAL model
• Shows lower bounds for edge coloring with Δ+k colors