Reductions in local certification

TL;DR

This paper introduces a framework for transferring lower bounds on certificate sizes in local certification problems, enabling polynomial lower bounds for various properties through hardness reductions.

Contribution

It proposes a notion of local hardness reduction in certification, allowing the transfer of lower bounds between properties, advancing understanding of local certification complexity.

Findings

Established a method to transfer lower bounds via local reductions.

Applied reductions to obtain polynomial lower bounds for classical properties.

Enhanced the theoretical toolkit for analyzing local certification complexity.

Abstract

Local certification is a topic originating from distributed computing, where a prover tries to convince the vertices of a graph $G$ that $G$ satisfies some property $\mathcal{P}$. To convince the vertices, the prover gives a small piece of information, called certificate, to each vertex, and the vertices then decide whether the property $\mathcal{P}$ is satisfied by just looking at their certificate and the certificates of their neighbors. When studying a property $\mathcal{P}$ in the perspective of local certification, the aim is to find the optimal size of the certificates needed to certify $\mathcal{P}$, which can be viewed a measure of the local complexity of $\mathcal{P}$. A certification scheme is considered to be efficient if the size of the certificates is polylogarithmic in the number of vertices. While there have been a number of meta-theorems providing efficient certification schemes for general graph classes, the proofs of the lower bounds on the size of the certificates are usually very problem-dependent. In this work, we introduce a notion of hardness reduction in local certification, and show that we can transfer a lower bound on the certificates for a property $\mathcal{P}$ to a lower bound for another property $\mathcal{P}'$, via a (local) hardness reduction from $\mathcal{P}$ to $\mathcal{P}'$. We then give a number of applications in which we obtain polynomial lower bounds for many classical properties using such reductions.