Renaming in distributed certification

TL;DR

This paper explores methods to reduce identifier ranges in distributed network certification, enabling more efficient local verification of global properties with optimal certificate sizes.

Contribution

It introduces new techniques for renaming identifiers in local certification, including non-injective encoding with a global certificate, and demonstrates their applications to certify various graph properties efficiently.

Findings

Identifier renaming reduces certificate size in local certification.

Global certificates can encode compact, non-injective identifier assignments.

Optimal certificates of size O(n) are achieved for multiple graph properties.

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.