The $2$-connected bottleneck Steiner network problem is NP-hard in any $\ell_p$ plane

Abstract: Bottleneck Steiner networks model energy consumption in wireless ad-hoc networks. The task is to design a network spanning a given set of terminals and at most $k$ Steiner points such that the length of the longest edge is minimised. The problem has been extensively studied for the case where an optimal solution is a tree in the Euclidean plane. However, in order to model a wider range of applications, including fault-tolerant networks, it is necessary to consider multi-connectivity constraints for networks embedded in more general metrics. We show that the $2$-connected bottleneck Steiner network problem is NP-hard in any planar $p$-norm and, in fact, if P$\,\neq\,$NP then an optimal solution cannot be approximated to within a ratio of ${2}^\frac{1}{p}-\epsilon$ in polynomial time for any $\epsilon >0$ and $1\leq p< \infty$.