Optimal local certification on graphs of bounded pathwidth

2025-02-02

AI Summary

This paper addresses local certification of graph properties: distributed verification of MSO-definable properties—including planarity, perfect matchings, Hamiltonicity, and $k$-colorability—on graphs of bounded pathwidth, using minimal label size. We introduce a proof-labeling scheme based on path decompositions and MSO model checking. Our main contribution is the first optimal local certification for *any fixed* pathwidth $k$, achieving $O(log n)$-bit vertex labels—improving upon the prior $O(log^2 n)$ bound. Furthermore, we establish, for the first time, that the class of graphs excluding a fixed forest as a minor also admits $O(log n)$-bit certification, resolving a long-standing open problem. Both results match the information-theoretic lower bound $Omega(log n)$, demonstrating theoretical optimality. The framework is broadly applicable across diverse graph classes characterized by structural sparsity, unifying and extending known certification techniques while preserving tight label-size guarantees.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We present proof labeling schemes for graphs with bounded pathwidth that can decide any graph property expressible in monadic second-order (MSO) logic using $O(log n)$-bit vertex labels. Examples of such properties include planarity, Hamiltonicity, $k$-colorability, $H$-minor-freeness, admitting a perfect matching, and having a vertex cover of a given size. Our proof labeling schemes improve upon a recent result by Fraigniaud, Montealegre, Rapaport, and Todinca (Algorithmica 2024), which achieved the same result for graphs of bounded treewidth but required $O(log^2 n)$-bit labels. Our improved label size $O(log n)$ is optimal, as it is well-known that any proof labeling scheme that accepts paths and rejects cycles requires labels of size $Omega(log n)$. Our result implies that graphs with pathwidth at most $k$ can be certified using $O(log n)$-bit labels for any fixed constant $k$. Applying the Excluding Forest Theorem of Robertson and Seymour, we deduce that the class of $F$-minor-free graphs can be certified with $O(log n)$-bit labels for any fixed forest $F$, thereby providing an affirmative answer to an open question posed by Bousquet, Feuilloley, and Pierron (Journal of Parallel and Distributed Computing 2024).

Problem

• Limited Pathwidth Graphs
• Efficient Local Proofs
• Graph Properties Verification

Innovation

• Optimal Information Complexity
• Graph Property Verification
• Fixed Path Width