Short Graph Sketches Suffice for Error-resilient Leader Verification in CONGEST

Locally Checkable Proofs (LCPs) enable the verification of global graph properties using locally checkable certificates assigned by a prover. Recently, this framework was extended to Locally Checkable Proofs-with-Errors (LCPE), where an adversary may corrupt some certificates. Existing LCPE algorithms, however, are designed for the LOCAL model, whose unbounded communication makes them unsuitable for direct implementation in the bandwidth-restricted CONGEST model. We initiate the study of efficient CONGEST implementations of LCPE through the \textsc{unique-leader} verification problem on trees. The main challenge is that tolerating $\varepsilon$ certificate errors requires each node to reason about its $(2\varepsilon+1)$-hop neighborhood, whose exact topology may require up to $O(\Delta^{2\varepsilon+1}\log n)$ bits to communicate. To overcome this bottleneck, we introduce \emph{local graph sketches}, together with the notions of \emph{imagined trees} and \emph{imagined certifications}, which encode precisely the information needed for verification using only $O(\varepsilon^2\log n)$ bits per node. Based on these sketches, we design an LCPE algorithm that tolerates up to $\varepsilon$ adversarial certificate errors and constructs the required sketches in $O(\varepsilon^2)$ communication rounds in the CONGEST model. We complement our algorithm with a matching impossibility result: even in the strictly more powerful LOCAL model, and even with unbounded certificate size, no verification scheme with view distance at most $\varepsilon$ can tolerate $\varepsilon$ adversarial certificate errors. Since every CONGEST algorithm can be simulated in LOCAL, this lower bound immediately applies to CONGEST, showing that a view distance exceeding $\varepsilon$ is unavoidable.