Decreasing verification radius in local certification

TL;DR

This paper explores reducing the local verification radius in distributed graph certification by encoding neighborhoods into certificates, balancing radius reduction with certificate size, and establishing near-optimal bounds.

Contribution

It introduces a method to decrease the verification radius in local certification by encoding neighborhoods, along with a lower bound showing near-optimality of this approach.

Findings

Procedure to decrease verification radius by encoding neighborhoods

Lower bound on certificate size increase for radius reduction

Approach is close to optimal in trade-off between radius and certificate size

Abstract

This paper deals with local certification, specifically locally checkable proofs: given a graph property, the task is to certify whether a graph satisfies the property. The verification of this certification needs to be done locally without the knowledge of the whole graph. More precisely, a distributed algorithm, called a verifier, is executed on each vertex. The verifier observes the local neighborhood up to a constant distance and either accepts or rejects. We examine the trade-off between the visibility radius and the size of certificates. We describe a procedure that decreases the radius by encoding the neighbourhood of each vertex into its certificate. We also provide a corresponding lower bound on the required certificate size increase, showing that such an approach is close to optimal.