Renaming in distributed certification

2024-09-23

arXiv.org

AI Summary

This work addresses identifier compression in distributed local certification: minimizing the numeric range of node identifiers—while relying solely on local verification—such that global graph properties remain certifiable. We propose a systematic identifier renaming framework covering both controllable and uncontrollable assignment settings, and introduce a globally certificate-driven compact non-injective encoding scheme. For fundamental graph properties—including bounded clique number (≤ ℓ), bounded diameter (≤ ℓ), independence number ≤ 2, and bipartiteness—we construct the first local or global certification schemes of size O(n), all matching the information-theoretic lower bounds. These results constitute the currently optimal solutions for identifier compression in local certification.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Local Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 certificates of size $O(n)$. All these results are optimal.

Problem

• Reduce identifier range in local certification
• Enable compact global certificate encoding
• Optimize certification for network properties

Innovation

• Reduces identifier range in local certification
• Renames identifiers with or without choice
• Encodes compact non-injective identifier assignments