To achieve fast recovery from link failures, most modern communication networks feature fully decentralized fast re-routing mechanisms. These re-routing mechanisms rely on pre-installed static re-routing rules at the nodes (the routers), which depend only on local failure information, namely on the failed links incident to the node. Ideally, a network is perfectly resilient: the re-routing rules ensure that packets are always successfully routed to their destinations as long as the source and the destination are still physically connected in the underlying network after the failures. Unfortunately, there are examples where achieving perfect resilience is not possible. Surprisingly, only very little is known about the algorithmic aspect of when and how perfect resilience can be achieved. We investigate the computational complexity of analyzing such local fast re-routing mechanisms. Our main result is a negative one: we show that even checking whether a given set of static re-routing rules ensures perfect resilience is coNP-complete. We also show coNP-completeness of the so-called ideal resilience, a weaker notion of resilience often considered in the literature. Additionally, we investigate other fundamental variations of the problem. In particular, we show that our coNP-completeness proof also applies to scenarios where the re-routing rules have specific patterns (known as skipping in the literature). On the positive side, for scenarios where nodes do not have information about the link from which a packet arrived (the so-called in-port), we present linear-time algorithms for both the verification and synthesis problem for perfect resilience.

翻译：为实现链路故障的快速恢复，大多数现代通信网络均采用完全分布式的快速重路由机制。这些重路由机制依赖于节点（路由器）上预装的静态重路由规则，这些规则仅依据本地故障信息（即节点关联的故障链路）进行决策。理想情况下，网络应具备完美弹性：只要故障后源节点与目的节点在底层网络中仍保持物理连通，重路由规则就能确保数据包始终成功路由至目的地。然而，存在某些场景下无法实现完美弹性的实例。令人惊讶的是，关于何时及如何实现完美弹性的算法特性，目前所知甚少。本文研究了分析此类本地快速重路由机制的计算复杂性。我们的核心结论是否定性的：即使仅验证给定静态重路由规则集是否能确保完美弹性，该问题也是coNP完全的。我们还证明了文献中常考虑的较弱弹性概念——即所谓理想弹性——的验证问题同样是coNP完全的。此外，我们探究了该问题的其他基本变体。特别地，我们证明了当重路由规则具有特定模式（文献中称为"跳过"模式）时，我们的coNP完全性结论依然成立。在积极方面，针对节点无法获知数据包到达链路（即所谓入端口）信息的场景，我们提出了适用于完美弹性验证与综合问题的线性时间算法。