局部决策的层次结构
A hierarchy of local decision
摘要：
受所谓分布式图自动机最近结果的启发，我们在分布式网络计算的框架中扩展了分布式决策的概念。我们表明，通过使用分布式决策机制之间的相互作用的证明者和disprover，证书的大小分发到节点证明一个给定的网络属性可以大大减少。例如，我们证明了最小生成树可以认证$n$ -节点图中的$O(\log n)$位证书，只有一个之间的相互作用的证明者和disprover，而它是已知的，认证MST需要$Ω(\log^2 n)$位证书，如果只有证明者可以采取行动。对于一些简单的图形属性，这种改进甚至可以是指数级的。例如，已知如果只有证明者可以动作，则证明非平凡自同构的存在需要$Ω(n^2)$位。我们发现，有一个协议，证明者和disprover之间的两个相互作用，使证明非平凡的自同构与$O(\log n)$位证书。这些结果是通过定义和分析一个本地层次的决策，推广了经典的概念证明标签计划和本地可检查的证明。
英文摘要：
We extend the notion of distributed decision in the framework of distributed network computing, inspired by recent results on so-called distributed graph automata. We show that, by using distributed decision mechanisms based on the interaction between a prover and a disprover, the size of the certificates distributed to the nodes for certifying a given network property can be drastically reduced. For instance, we prove that minimum spanning tree can be certified with $O(\log n)$-bit certificates in $n$-node graphs, with just one interaction between the prover and the disprover, while it is known that certifying MST requires $Ω(\log^2 n)$-bit certificates if only the prover can act. The improvement can even be exponential for some simple graph properties. For instance, it is known that certifying the existence of a nontrivial automorphism requires $Ω(n^2)$ bits if only the prover can act. We show that there is a protocol with two interactions between the prover and the disprover enabling to certify nontrivial automorphism with $O(\log n)$-bit certificates. These results are achieved by defining and analysing a local hierarchy of decision which generalizes the classical notions of proof-labelling schemes and locally checkable proofs.
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