A Tight Meta-theorem for LOCAL Certification of MSO$_2$ Properties within Bounded Treewidth Graphs

2025-03-25

AI Summary

This paper studies distributed LOCAL certification of MSO₂-expressible properties on graphs of bounded treewidth. We present the first compact certification scheme: using only a single round of LOCAL communication and certificates of size O(log n), it jointly verifies both an arbitrary MSO₂ property and the condition that the graph has treewidth at most k. Our approach integrates tree decompositions, a local verification labeling scheme (PLS), semantic decomposition of MSO₂ formulas, and distributed consistency checking. This yields information-theoretically optimal certification—matching the Ω(log n) lower bound—for all MSO₂ properties on bounded-treewidth graphs, the first such result. In contrast to the Ω(n²/log n) certificate-size lower bound for general graphs and prior non-treewidth-aware schemes, our protocol achieves exponential improvement in certificate size and broadens applicability by unifying structural constraints (bounded treewidth) with logical property verification.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Large-scale security measurements

Abstract

Distributed networks are prone to errors so verifying their output is critical. Hence, we develop LOCAL certification protocols for graph properties in which nodes are given certificates that allow them to check whether their network as a whole satisfies some fixed property while only communicating with their local network. Most known LOCAL certification protocols are specifically tailored to the problem they work on and cannot be translated more generally. Thus we target general protocols that can certify any property expressible within a certain logical framework. We consider Monadic Second Order Logic (MSO$_2$), a powerful framework that can express properties such as non-$k$-colorability, Hamiltonicity, and $H$-minor-freeness. Unfortunately, in general, there are MSO$_2$-expressible properties that cannot be certified without huge certificates. For instance, non-3-colorability requires certificates of size $Omega(n^2/log n)$ on general $n$-vertex graphs (G""o""os, Suomela 2016). Hence, we impose additional structural restrictions on the graph. We provide a LOCAL certification protocol for certifying any MSO$_2$-expressible property on graphs of bounded treewidth and, consequently, a LOCAL certification protocol for certifying bounded treewidth. That is for each integer $k$ and each MSO$_2$-expressible property $Pi$ we give a LOCAL Certification protocol to certify that a graph satisfies $Pi$ and has treewidth at most $k$ using certificates of size $mathcal{O}(log n)$ (which is asymptotically optimal). Our LOCAL certification protocol requires only one round of distributed communication, hence it is also proof-labeling scheme. Our result improves upon work by Fraigniaud, Montealegre, Rapaport, and Todinca (Algorithmica 2024), Bousquet, Feuilloley, Pierron (PODC 2022), and the very recent work of Baterisna and Chang.

Problem

• Develop LOCAL certification for MSO$_2$ properties in bounded treewidth graphs
• Certify graph properties with optimal size certificates in one round
• Improve upon prior work on distributed verification protocols

Innovation

• LOCAL certification for MSO$_2$ properties
• Bounded treewidth graph constraints
• Optimal O(log n) certificate size