Proving there is a leader without naming it

TL;DR

This paper investigates whether leader certification can be achieved without node identifiers in various graph classes, showing that structural properties influence the complexity of local certification.

Contribution

It demonstrates that in certain graph classes, leader certification can be done with sublogarithmic certificates without using identifiers, highlighting the impact of graph structure.

Findings

Leader certification can be achieved with sublogarithmic certificates in specific graph classes.

Identifiers may not be necessary for leader certification in certain topologies.

Structural properties of graphs significantly influence local certification complexity.

Abstract

Local certification is a mechanism for certifying to the nodes of a network that a certain property holds. In this framework, nodes are assigned labels, called certificates, which are supposed to prove that the property holds. The nodes then communicate with their neighbors to verify the correctness of these certificates. Certifying that there is a unique leader in a network is one of the most classical problems in this setting. It is well-known that this can be done using certificates that encode node identifiers and distances in the graph. These require $O(\log n)$ and $O(\log D)$ bits respectively, where $n$ is the number of nodes and $D$ is the diameter. A matching lower bound is known in cycle graphs (where $n$ and $D$ are equal up to multiplicative constants). A recent line of work has shown that network structure greatly influences local certification. For example, certifying that a network does not contain triangles takes $\Theta(n)$ bits in general graphs, but only $O(\log n)$ bits in graphs of bounded treewidth. This observation raises the question: Is it possible to achieve sublogarithmic leader certification in graph classes that do not contain cycle graphs? And since in that case we cannot write identifiers in a certificate, do we actually need identifiers at all in such topologies? [We answer these questions with results on small diameter graphs, chordal graphs, grids, and dense graphs. See full abstract in the paper.]