Error-resilient Distributed Local Verification

TL;DR

This paper introduces new methods for verifying graph properties in distributed networks, including a novel cycle detection scheme with fewer labels and an algorithmic framework for error-tolerant verification under adversarial label corruption.

Contribution

It presents a new cycle detection gadget using only 2 labels and a general framework for error-resilient verification in distributed settings.

Findings

A 2-label cycle detection scheme with 3-hop neighborhood access.

A framework transforming verification algorithms to tolerate multiple label errors.

Lower bounds relating error tolerance to neighborhood size.

Abstract

We study verification (decision) problems for graph properties in distributed networks under the locally checkable labeling framework, where nodes use labels (proofs) and local neighborhoods to decide acceptance or rejection. Our focus is twofold. First, we study cycle detection. While it is known that this can be verified using 3 labels with access to the 1-hop neighborhood, we introduce a novel gadget that encodes direction along a path using only 2 labels and access to a 3-hop neighborhood. This yields a cycle-detection labeling scheme with just 2 labels and may be of independent interest. Second, we consider adversarially corrupted labelings, where each node has access to a local neighborhood within which a fraction of nodes may receive erroneous labels. We introduce a general algorithmic framework, called refix, that transforms a base verification algorithm for a property P operating on labels within a d-hop neighborhood into one that tolerates up to i erroneous labels within a radius d+2i, by accessing a d+2i-hop neighborhood. We demonstrate applications to cycle detection, cycle absence, and bipartiteness, and provide lower bounds relating the number of errors to the required neighborhood size.