🤖 AI Summary
This work establishes lower bounds on certificate size for local certification of distributed graph properties. Focusing on classical properties—including connectivity, matching, and coloring—it introduces the first local hardness reduction framework. By constructing local graph gadgets and applying information-theoretic analysis within the local model, the framework systematically transfers certificate-size lower bounds from one class of properties to another, overcoming the limitations of property-specific proofs. This approach achieves the first transferable lower-bound results in local certification. It uniformly establishes polynomial-scale (Ω(n)) lower bounds for multiple fundamental properties, exposing their inherent local complexity. The framework provides a general tool for local certification theory and advances the understanding of the fundamental limits of distributed verification efficiency.
📝 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.