AI 中文总结
研究在CONGEST模型中通过树上唯一领导者验证问题实现LCPE的高效实现。核心方法是引入局部图草图等概念，每个节点用少量位编码验证信息，设计出可容忍错误的算法，并给出匹配的不可能性结果。
AI 中文摘要
局部可验证证明（LCP）允许使用证明者分配的局部可验证证书来验证全局图属性。最近，该框架扩展到了带有错误的局部可验证证明（LCPE），其中对手可能会破坏一些证书。然而，现有的LCPE算法是为LOCAL模型设计的，其无界通信使其不适用于带宽受限的CONGEST模型。我们通过树上的唯一领导者验证问题启动了对LCPE在CONGEST中高效实现的研究。主要挑战在于容忍ε个证书错误需要每个节点考虑其（2ε + 1）跳邻域，其确切拓扑可能需要多达O（Δ ^ （2ε + 1）log n）位来通信。为克服此瓶颈，我们引入了局部图草图，以及想象树和想象认证的概念，每个节点仅使用O（ε ^ 2 log n）位就精确编码了验证所需的信息。基于这些草图，我们设计了一种LCPE算法，该算法可容忍多达ε个对抗性证书错误，并在CONGEST模型中以O（ε ^ 2）通信轮次构建所需草图。我们用一个匹配的不可能性结果补充了我们的算法：即使在更强大的LOCAL模型中，即使证书大小无界，视图距离最多为ε的验证方案也无法容忍ε个对抗性证书错误。由于每个CONGEST算法都可以在LOCAL中模拟，因此该下限立即适用于CONGEST，表明视图距离超过ε是不可避免的。
英文摘要
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 $O(Δ^{2\varepsilon+1}\log n)$ 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.
