Complexity of Perfect and Ideal Resilience Verification in Fast Re-Route Networks

TL;DR

This work investigates the algorithmic complexity of configuring local fast re-routing in networks to achieve perfect or ideal resilience. It proves that verifying a given local forwarding pattern for perfect resilience is coNP-complete, and that verifying ideal resilience is also coNP-complete, with these hardness results persisting even on planar graphs; by contrast, synthesis of perfect resilience is achievable in polynomial time, and linear-time algorithms exist for in-port oblivious routing variants. A sharp, tractable criterion is provided for in-port oblivious routings: a perfectly resilient in-port oblivious routing exists if and only if every simple cycle in the graph has length at most $3$, enabling linear-time verification and construction. The paper also shows that the in-port oblivious setting yields significant memory savings and practical verifiability, while highlighting open questions about constructing resilient routing patterns in general, and discussing ETH-based lower bounds on the problem size. Overall, the results delineate a clear boundary between hardness and tractable cases in resilience verification and synthesis, guiding future design of robust, decentralized routing schemes.

Abstract

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.

Figures (6)

• Figure 1: Example of a graph with target $t=v_5$ and a perfectly resilient skipping forwarding pattern. For the sake of readability, we represent (here and in the following figures) links incident to a node $v$ by only stating the other endpoint of the link. The routing determined by this forwarding pattern, failure scenario $F=\{\{v_2,v_5\},\{v_3,v_5\}\}$, and the starting node $s=v_1$ is ${\pi=(v_1,v_2,v_1,v_3,v_1,v_4,v_5)}$.
• Figure 2: A triangle consisting of three nodes $s,v,$ and $u$, where $u$ is the unique node where all paths from either $s$ or $v$ to $t$ pass through $u$ and $u'$ is the second node on the shortest path from $u$ to $t$.
• Figure 3: A graph and a perfectly resilient in-port oblivious skipping forwarding pattern for the target $t$.
• Figure 4: Example of the construction of an instance in the proof of \ref{['thm:perfecthard']}.
• Figure 5: Construction of the graph in the proof of \ref{['thm:idealhard']} for the formula $\phi = (x_1 \lor x_2 \lor \neg x_4) \land (\neg x_1 \lor \neg x_3 \lor x_4) \land (x_1 \lor \neg x_2 \lor x_3)$excluding the clique $K$ and the target $t$.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Definition 7
• Definition 8
• Theorem 2
• proof