A Meta-Theorem for Distributed Certification

TL;DR

This paper introduces a meta-theorem demonstrating that all graph properties definable in MSO logic can be efficiently certified in distributed systems with small certificates and a single communication round, specifically for graphs of bounded treewidth.

Contribution

It establishes a universal certification mechanism for MSO-definable graph properties in distributed systems with bounded treewidth, using small certificates and minimal communication.

Findings

Certificates of size O(log^2 n) suffice for MSO-definable properties

Verification requires only one round of neighbor communication

Applicable to graphs with bounded treewidth

Abstract

Distributed certification, whether it be proof-labeling schemes, locally checkable proofs, etc., deals with the issue of certifying the legality of a distributed system with respect to a given boolean predicate. A certificate is assigned to each process in the system by a non-trustable oracle, and the processes are in charge of verifying these certificates, so that two properties are satisfied: completeness, i.e., for every legal instance, there is a certificate assignment leading all processes to accept, and soundness, i.e., for every illegal instance, and for every certificate assignment, at least one process rejects. The verification of the certificates must be fast, and the certificates themselves must be small. A large quantity of results have been produced in this framework, each aiming at designing a distributed certification mechanism for specific boolean predicates. This paper presents a "meta-theorem", applying to many boolean predicates at once. Specifically, we prove that, for every boolean predicate on graphs definable in the monadic second-order (MSO) logic of graphs, there exists a distributed certification mechanism using certificates on $O(\log^2n)$ bits in $n$-node graphs of bounded treewidth, with a verification protocol involving a single round of communication between neighbors.