Proving there is a leader without naming it

TL;DR

This work analyzes how graph topology and identifier constraints shape the local certification of a unique leader. It extends the Göös-Suomela lower-bound framework to derive an $\Omega(\log n)$ barrier in graphs of bounded diameter, and simultaneously constructs sublogarithmic upper bounds in structured anonymous graphs (chordal and grids) as well as in dense graphs and in scenarios with small identifier ranges. Key contributions include a $2$-local $O(\log D)$ scheme for anonymous chordal graphs, a $1$-local $O(\log D)$ scheme for anonymous grids, an $O(\log \log n)$ upper bound in dense graphs, and $O(\log D)$ upper bounds when a small root identifier exists. The results illuminate how holes, diameter, sparsity/density, and identifier ranges interact to determine the feasibility of sublogarithmic certification, and they raise open questions about robust hole notions and complete properties under local reductions.

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 $Θ(n)$ 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.]

Figures (7)

• Figure 1: Left picture: illustration of the type of graphs for which the Göös-Suomela proof works, with a hole and two thin parts (the graph is the gray part). Middle picture: a chordal graph. Right picture: a 2-by-3 grid.
• Figure 2: An example of a graph $G(a,b)$ consisting of $a$ and $b$.
• Figure 3: An example of a graph created by connecting two yes-instances $G(a_i,b_i)$ and $G(a_j,b_j)$.
• Figure 4: Visualization of a case that R2 prevents.
• Figure 5: Visualization of a case that R3 prevents. Note that even if there would be an edge between $u$ and $z$ the cycle would still be too long.

Theorems & Definitions (22)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 5
• proof
• proof
• proof
• Theorem 7