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.

翻译：我们在\textit{局部可检查证明}（LCPs）框架下，研究分布式网络中图性质的局部验证。在LCP中，证明者为节点分配证明标签，若图满足该性质，分布式验证器必须使所有节点接受；否则至少有一个节点拒绝。每个节点基于局部邻域（称为\textit{视距}）做出决策。我们关注两方面：首先，研究环的存在性（即图是否包含环，而非无环性）。我们证明环存在性仅需$3$个证明标签和视距$1$即可验证，并建立了匹配的下界。更重要的是，受基于BFS距离的方向编码技术启发，我们引入一种新型结构，通过重复出现字符串$001101$，仅用$2$个标签和视距$3$编码方向。尽管该结构是为环存在性而设计，但可能对其他验证任务有用。其次，我们提出一种\textit{错误证明}模型，其中攻击者可能破坏每个节点$(2i+1)$跳邻域内至多$i$个节点的证明标签。我们提出一种名为\textbf{\texttt{refix}}的算法框架，可将无错误验证器转化为容忍此类错误的验证器，代价是视距增加至$2i+1$。我们在环存在性、无环性和二部性问题上演示该框架，并建立了错误数量与所需视距之间的下界关系。最后，我们证明环存在性的$2$标签、视距$3$验证器可在\textsc{CONGEST}模型中用$3$轮实现，为在通信约束下实现LCPs迈出了第一步。