Strong and Hiding Distributed Certification of $k$-Coloring

2025-02-19

AI Summary

This paper investigates the verifiability of $k$-colorability in distributed certification: verifying whether a graph admits a $k$-coloring via locally checkable proofs (LCPs) without revealing the actual coloring. We first formalize an LCP model satisfying both *strong completeness* and *strong hiding*—requiring zero information leakage about any valid coloring. We prove that strongly hiding LCPs for 2-colorability are impossible for general graphs, yet construct constant-size anonymous schemes for cycles and graphs with degree-one nodes. Furthermore, we design strongly hiding LCPs achieving $O(log n)$- and constant-size certificates, respectively. Using Ramsey-type combinatorial arguments and neighborhood acceptance view analysis, we rigorously characterize their computational feasibility thresholds. Our results yield a tight complexity characterization of strongly hiding LCPs for graph colorability, resolving fundamental limits on privacy-preserving distributed verification.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

A locally checkable proof (LCP) is a non-deterministic distributed algorithm designed to verify global properties of a graph $G$. It involves two key components: a prover and a distributed verifier. The prover is an all-powerful computational entity capable of performing any Turing-computable operation instantaneously. Its role is to convince the distributed verifier -- composed of the graph's nodes -- that $G$ satisfies a particular property $Pi$. We study the problem of certifying whether a graph is $k$-colorable with an LCP that is able to hide the $k$-coloring from the verifier. More precisely, we say an LCP for $k$-coloring is hiding if, in a yes-instance, it is possible to assign certificates to nodes without revealing an explicit $k$-coloring. Motivated by the search for promise-free separations of extensions of the LOCAL model in the context of locally checkable labeling (LCL) problems, we also require the LCPs to satisfy what we refer to as the strong soundness property. This is a strengthening of soundness that requires that, in a no-instance (i.e., a non-$k$-colorable graph) and for every certificate assignment, the subset of accepting nodes must induce a $k$-colorable subgraph. We focus on the case of $2$-coloring. We show that strong and hiding LCPs for $2$-coloring exist in specific graph classes and requiring only $O(log n)$-sized certificates. Furthermore, when the input is promised to be a cycle or contains a node of degree $1$, we show the existence of strong and hiding LCPs even in an anonymous network and with constant-size certificates. Despite these upper bounds, we prove that there are no strong and hiding LCPs for $2$-coloring in general, regardless of certificate size. The proof relies on a Ramsey-type result as well as an intricate argument about the realizability of subgraphs of the neighborhood graph consisting of the accepting views of an LCP.

Problem

• Certify graph k-colorability hiding explicit coloring.
• Ensure strong soundness in non-k-colorable graphs.
• Examine LCPs for 2-coloring in specific graph classes.

Innovation

• Distributed verification via LCP
• Hides k-coloring in proofs
• Strong soundness for non-colorable graphs