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Decreasing verification radius in local certification

Laurent Feuilloley, Jan Janoušek, Jan Matyáš Křišťan, Josef Erik Sedláček

TL;DR

This work investigates local certification through locally checkable proofs in distributed graphs, focusing on how to reduce the verification radius without sacrificing correctness. It introduces a general packing technique that encodes the $\delta$-neighborhood into certificates via packets, achieving a certificate-size blow-up of $\mathcal{O}((\Delta-1)^{\delta}(\Delta \log(n) + s(n) + \ell(n)))$ for any $\delta<r$, while preserving completeness and soundness. The authors also establish a lower bound showing that some certificate-size growth proportional to $C(\Delta-1)^{\delta-1}$ is necessary to decrease the radius, and discuss cases where the logarithmic factors become additive. Overall, the results provide a versatile framework for trading communication radius against per-node memory and computation in distributed verification, with implications for self-stabilization and graph property testing.

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.

Decreasing verification radius in local certification

TL;DR

This work investigates local certification through locally checkable proofs in distributed graphs, focusing on how to reduce the verification radius without sacrificing correctness. It introduces a general packing technique that encodes the -neighborhood into certificates via packets, achieving a certificate-size blow-up of for any , while preserving completeness and soundness. The authors also establish a lower bound showing that some certificate-size growth proportional to is necessary to decrease the radius, and discuss cases where the logarithmic factors become additive. Overall, the results provide a versatile framework for trading communication radius against per-node memory and computation in distributed verification, with implications for self-stabilization and graph property testing.

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.
Paper Structure (7 sections, 1 theorem, 1 equation)

This paper contains 7 sections, 1 theorem, 1 equation.

Table of Contents

  1. Introduction
  2. Previous work
  3. Our contribution
  4. Preliminaries
  5. Decreasing the radius of a proof labeling scheme
  6. Overview of the proof technique
  7. Encoding neighborhood into certificates

Key Result

theorem thmcountertheorem

Given an $r$-local proof labeling scheme $(f_r, \mathcal{A}_r)$ of size $s$ certifying a graph property $\mathcal{P}$, for every $\delta < r$, we can construct an $(r - \delta)$-local proof labeling scheme certifying $\mathcal{P}$ with certificates of size where $\ell(n)$ is the maximum size of a label and $\Delta \geq 3$ is the maximum degree of the input graph.

Theorems & Definitions (3)

  • theorem thmcountertheorem
  • definition thmcounterdefinition
  • definition thmcounterdefinition