The final answer to the complexity of a basic problem in resilient network design

To provide resilience to failures of the multi-commodity flow network, either in the failure-free state flows can be routed along multiple paths and over-dimensioned, or whenever a failure occurs flows can be restored along unaffected paths. The complexity of the network design depends on the selected method of providing resilience and on a number of design options—whether single or multiple commodities and single- or multi-element failures are considered, if the reaction to failures is dependent or independent on the failure, which mechanism of capacity release and reuse is applied, etc. For almost all combinations of those choices either the corresponding design problem has already been shown to be NP-hard or a compact linear programming formulation of the problem has been provided. The only case that has resisted an answer is when flows are restored in a state-dependent manner using the stub release mechanism. In this paper it is proved that the corresponding network design problem is NP-hard even for a single commodity and for single-element failures. The proof is based on the reduction of the Hamiltonian path problem.