Distributed Local Verification using Proofs with(out) Errors
We study local verification of graph properties in distributed networks under the framework of \emph{locally checkable proofs} (LCPs). In an LCP, a prover assigns proof labels to nodes, and a distributed verifier must make all nodes accept if the graph satisfies the property, while at least one node rejects otherwise. Each node bases its decision on a local neighborhood, called its \emph{view distance}. Our focus is twofold. First, we study cycle existence, i.e., whether a graph contains a cycle (as opposed to cycle-freeness). We show that cycle existence admits verification with only $3$ proof labels and view distance $1$, and establish a matching lower bound. More importantly, inspired by direction-encoding techniques based on BFS distances, we introduce a novel gadget that encodes direction using only $2$ labels and view distance $3$ through repeated occurrences of the string $001101$. Although developed for cycle existence, this gadget may be useful for other verification tasks. Second, we introduce an \emph{erroneous proof} model in which an adversary may corrupt proof labels of at most $i$ nodes within the $(2i+1)$-hop neighborhood of each node. We present an algorithmic framework, called \textbf{\texttt{refix}}, that transforms an error-free verifier into one that tolerates such errors at the cost of a view distance of $2i+1$. We demonstrate the framework on cycle existence, cycle-freeness, and bipartiteness, and establish lower bounds relating the number of errors to the required view distance. Finally, we show that our $2$-label, view-distance-$3$ verifier for cycle existence admits a $3$-round implementation in the \textsc{CONGEST} model, providing a first step toward implementing LCPs under communication constraints.