亏格有界图的局部证明
Local Certification of Graphs with Bounded Genus
摘要：
Naor，Parter和Yogev [SODA 2020]最近设计了一个编译器，用于自动将标准的集中式交互协议转换为分布式交互协议，如Kol，Oshman和Saxena [PODC 2018]所介绍的那样。特别地，通过使用该编译器，用于确定某个固定图类的成员资格的每个线性时间算法可以被翻译成用于该类的$\mathsf{dMAM}(O(\log n))$协议，即，在$n$节点图中具有$O(\log n)$比特证明大小的分布式交互协议，以及在$n$节点图之间的三个交互。（中心化）计算无界但不可信的证明者Merlin和（分散化）随机计算有限的验证者Arthur。作为一个推论，有一个$\mathsf{dMAM}(O(\log n))$协议的类的平面图，以及类的图有界亏格。我们发现，存在一个分布式的交互式协议的类图有界属执行一个单一的互动，从证明者到验证者，但保持证明大小的$O(\log n)$位。这个结果对于具有有界半亏格的图类（即可以嵌入到有界亏格的不可定向曲面上的图）也成立。本文中描述的交互式协议实际上是证明标记方案，即，交互式协议的一个子类，先前由Korman、Kutten和Peleg介绍[PODC 2005]。特别地，这些方案不需要验证者的任何随机化，并且证明通常可以由节点本身以低成本先验地计算。因此，我们的结果扩展了Feuilloley等人[PODC 2020]最近的平面图的证明标记方案，有界亏格的图，以及有界亏格的图。
英文摘要：
Naor, Parter, and Yogev [SODA 2020] recently designed a compiler for automatically translating standard centralized interactive protocols to distributed interactive protocols, as introduced by Kol, Oshman, and Saxena [PODC 2018]. In particular, by using this compiler, every linear-time algorithm for deciding the membership to some fixed graph class can be translated into a $\mathsf{dMAM}(O(\log n))$ protocol for this class, that is, a distributed interactive protocol with $O(\log n)$-bit proof size in $n$-node graphs, and three interactions between the (centralizer) computationally-unbounded but non-trustable prover Merlin, and the (decentralized) randomized computationally-limited verifier Arthur. As a corollary, there is a $\mathsf{dMAM}(O(\log n))$ protocol for the class of planar graphs, as well as for the class of graphs with bounded genus. We show that there exists a distributed interactive protocol for the class of graphs with bounded genus performing just a single interaction, from the prover to the verifier, yet preserving proof size of $O(\log n)$ bits. This result also holds for the class of graphs with bounded demi-genus, that is, graphs that can be embedded on a non-orientable surface of bounded genus. The interactive protocols described in this paper are actually proof-labeling schemes, i.e., a subclass of interactive protocols, previously introduced by Korman, Kutten, and Peleg [PODC 2005]. In particular, these schemes do not require any randomization from the verifier, and the proofs may often be computed a priori, at low cost, by the nodes themselves. Our results thus extend the recent proof-labeling scheme for planar graphs by Feuilloley et al. [PODC 2020], to graphs of bounded genus, and to graphs of bounded demigenus.
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