Renaming in distributed certification

TL;DR

The paper advances the theory of distributed local certification by showing how to compress and certify identifier renamings, both when the new identifiers can be chosen and when they cannot, and by linking renaming with global certificates through perfect hashing. It provides two main renaming schemes with tight certificate-size bounds and demonstrates broad applications, including optimal O(n)-bit certificates for K_l-freeness, bounded-diameter graphs, and independence-number constraints, as well as a global O(n) certificate for bipartiteness and a hashing-based approach for H-homomorphism. The results bridge renaming techniques, universal graphs, and hashing to achieve near-optimal space complexity in local and global certification settings. These contributions yield practical implications for efficiently certifying classic graph properties in distributed networks while maintaining provable optimality under common identifier-range assumptions.

Abstract

Local certification is the area of distributed network computing asking the following question: How to certify to the nodes of a network that a global property holds, if they are limited to a local verification? In this area, it is often essential to have identifiers, that is, unique integers assigned to the nodes. In this short paper, we show how to reduce the range of the identifiers, in three different settings. More precisely, we show how to rename identifiers in the classical local certification setting, when we can (resp.\ cannot) choose the new identifiers, and we show how a global certificate can help to encode very compactly a new identifier assignment that is not injective in general, but still useful in applications. We conclude with a number of applications of these results: For every $\ell$, there are local certification schemes for the properties of having clique number at most $\ell$, having diameter at most $\ell$, and having independence number at most~2, with certificates of size $O(n)$. We also show that there is a global certification scheme for bipartiteness with a certificate of size $O(n)$. All these results are optimal.

Figures (1)

• Figure 1: Illustration of the renaming. The edges which are drawn here are those of $\mathcal{T}$. For each internal node, its children are written in increasing order of the original identifiers. The integer in every node $u$ corresponds to $\mathsf{Renaming}(u)$. The red integer near a node $u$ corresponds to $|\mathcal{T}_u|$. (The original identifiers are omitted.)

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: Local certification
• Definition 2.2: Global certification
• Definition 2.3
• Theorem 2.4
• Theorem 3.1
• proof
• Theorem 3.1
• proof