Naor M., Parter M., Yogev E.: (The power of distributed verifiers in interactive proofs. In: 31st ACM-SIAM symposium on discrete algorithms (SODA), pp 1096–115, 2020. https://doi.org/10.1137/1.9781611975994.67) have recently demonstrated the existence of a distributed interactive proof for planarity (i.e., for certifying that a network is planar), using a sophisticated generic technique for constructing distributed IP protocols based on sequential IP protocols. The interactive proof for planarity is based on a distributed certification of the correct execution of any given sequential linear-time algorithm for planarity testing. It involves three interactions between the prover and the randomized distributed verifier (i.e., it is a dMAM protocol), and uses small certificates, on \(O(\log n)\) bits in n-node networks. We show that a single interaction with the prover suffices, and randomization is unecessary, by providing an explicit description of a proof-labeling scheme for planarity, still using certificates on just \(O(\log n)\) bits. We also show that there are no proof-labeling schemes—in fact, even no locally checkable proofs—for planarity using certificates on \(o(\log n)\) bits.

Naor M.，Parter M.，Yogev E. ：（交互式证明中分布式验证程序的功能。在：2020年第31届ACM-SIAM离散算法研讨会（SODA），第1096–115页。https://doi.org /10.1137/1.9781611975994.67）最近证明了存在一种分布式的交互式平面性证明（即，用于证明网络是平面的），它使用一种复杂的通用技术来构造基于顺序IP协议的分布式IP协议。平面度的交互式证明基于对任何给定的顺序线性时间算法进行平面度测试的正确执行的分布式证明。它涉及证明方和随机分布的验证方之间的三种交互（即，它是dMAM协议），并在n个节点网络中的\（O（\ log n）\）位。通过提供对平面度的证明标签方案的明确描述，我们仍然仅使用\（O（\ log n）\）位上的证书，即可证明与证明者的单次交互就足够了，并且没有必要进行随机化。我们还表明，对于使用\（o（\ log n）\）位上的证书的平面度，没有证明标签方案-实际上，甚至没有本地可检查的证明。