The 2-connected bottleneck Steiner network problem is NP-hard in any ℓ_p plane

07/08/2019

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by   M Brazil, et al.

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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≠NP then an optimal solution cannot be approximated to within a ratio of 2^1/p-ϵ in polynomial time for any ϵ >0 and 1≤ p< ∞.