Complexity landscape for local certification

2025-05-27

AI Summary

This work investigates the space complexity of local certification for distributed graph algorithms—i.e., the minimal certificate size required by a prover to label nodes so that a verifier can locally check a graph property. Focusing on fundamental anonymous structures (paths, cycles, and trees), we establish the first rigorous characterizations of fundamental complexity gaps: a tight gap between $O(1)$ and $Theta(log log n)$ on paths; novel hierarchical separations including $Theta(log log n)$ versus $Theta(log n)$; and an extension to trees yielding an $O(1)$ versus $Theta(log log d)$ gap, where $d$ is the diameter. Methodologically, we introduce a novel analytical toolkit integrating automata theory, number theory, and combinatorial arguments. Our results formally confirm that $Theta(log log n)$ is the smallest nontrivial achievable certification complexity for natural graph properties, and identify exceptional certifiable cases admitting no complexity gap.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity

Abstract

An impressive recent line of work has charted the complexity landscape of distributed graph algorithms. For many settings, it has been determined which time complexities exist, and which do not (in the sense that no local problem could have an optimal algorithm with that complexity). In this paper, we initiate the study of the landscape for space complexity of distributed graph algorithms. More precisely, we focus on the local certification setting, where a prover assigns certificates to nodes to certify a property, and where the space complexity is measured by the size of the certificates. Already for anonymous paths and cycles, we unveil a surprising landscape: - There is a gap between complexity $O(1)$ and $Theta(log log n)$ in paths. This is the first gap established in local certification. - There exists a property that has complexity $Theta(log log n)$ in paths, a regime that was not known to exist for a natural property. - There is a gap between complexity $O(1)$ and $Theta(log n)$ in cycles, hence a gap that is exponentially larger than for paths. We then generalize our result for paths to the class of trees. Namely, we show that there is a gap between complexity $O(1)$ and $Theta(log log d)$ in trees, where $d$ is the diameter. We finally describe some settings where there are no gaps at all. To prove our results we develop a new toolkit, based on various results of automata theory and arithmetic, which is of independent interest.

Problem

• Study space complexity gaps in local certification for distributed graphs
• Identify gaps between O(1) and Θ(log log n) in paths and cycles
• Generalize complexity gaps to trees and explore gap-free settings

Innovation

• Studies space complexity in local certification
• Reveals gaps in complexity for paths and cycles
• Develops toolkit using automata theory and arithmetic