Distributed Certification for Classes of Dense Graphs

TL;DR

This paper extends distributed certification techniques to dense graphs by developing proof-labeling schemes with logarithmic squared certificate sizes for properties definable in monadic second-order logic, applicable to graphs with bounded clique-width.

Contribution

It introduces a new PLS construction for bounded clique-width graphs, broadening the scope from sparse to dense graphs in distributed certification.

Findings

Existence of PLS with O(log^2 n) bits for MSO_1 properties on bounded clique-width graphs.

Extension of meta-theorem from bounded tree-width to bounded clique-width graphs.

Applicable to dense graph classes, expanding the reach of distributed certification methods.

Abstract

A proof-labeling scheme (PLS) for a boolean predicate $\Pi$ on labeled graphs is a mechanism used for certifying the legality with respect to $\Pi$ of global network states in a distributed manner. In a PLS, a certificate is assigned to each processing node of the network, and the nodes are in charge of checking that the collection of certificates forms a global proof that the system is in a correct state, by exchanging the certificates once, between neighbors only. The main measure of complexity is the size of the certificates. Many PLSs have been designed for certifying specific predicates, including cycle-freeness, minimum-weight spanning tree, planarity, etc. In 2021, a breakthrough has been obtained, as a meta-theorem stating that a large set of properties have compact PLSs in a large class of networks. Namely, for every $\mathrm{MSO}_2$ property $\Pi$ on labeled graphs, there exists a PLS for $\Pi$ with $O(\log n)$-bit certificates for all graphs of bounded tree-depth. This result has been extended to the larger class of graphs with bounded {tree-width}, using certificates on $O(\log^2 n)$ bits. We extend this result even further, to the larger class of graphs with bounded clique-width, which, as opposed to the other two aforementioned classes, includes dense graphs. We show that, for every $\mathrm{MSO}_1$ property $\Pi$ on labeled graphs, there exists a PLS for $\Pi$ with $O(\log^2 n)$ bit certificates for all graphs of bounded clique-width.