Locally Restricted Proof Labeling Schemes (Full Version)

TL;DR

This paper explores locally restricted proof labeling schemes in distributed graph verification, introducing relaxations that allow for approximate correctness and property testing to overcome inherent limitations.

Contribution

It defines and analyzes locally restricted PLS models with relaxations, connecting them to approximation algorithms and property testing in distributed verification.

Findings

Introduces approximate proof labeling schemes (APLS) for distributed optimization problems.

Develops testing proof labeling schemes (TPLS) for configured graph families.

Shows relaxations enable verification in locally restricted models.

Abstract

Introduced by Korman, Kutten, and Peleg (PODC 2005), a proof labeling scheme (PLS) is a distributed verification system dedicated to evaluating if a given configured graph satisfies a certain property. It involves a centralized prover, whose role is to provide proof that a given configured graph is a yes-instance by means of assigning labels to the nodes, and a distributed verifier, whose role is to verify the validity of the given proof via local access to the assigned labels. In this paper, we introduce the notion of a locally restricted PLS in which the prover's power is restricted to that of a LOCAL algorithm with a polylogarithmic number of rounds. To circumvent inherent impossibilities of PLSs in the locally restricted setting, we turn to models that relax the correctness requirements by allowing the verifier to accept some no-instances as long as they are not "too far" from satisfying the property in question. To this end, we evaluate (1) distributed graph optimization problems (OptDGPs) based on the notion of an approximate proof labeling scheme (APLS) (analogous to the type of relaxation used in sequential approximation algorithms); and (2) configured graph families (CGFs) based on the notion of atesting proof labeling schemes (TPLS) (analogous to the type of relaxation used in property testing algorithms).