An exact algorithm for the bottleneck 2-connected $k$-Steiner network problem in $L_p$ planes

We present the first exact polynomial time algorithm for constructing optimal geometric bottleneck 2-connected Steiner networks containing at most $k$ Steiner points, where $k>2$ is a constant. Given a set of $n$ vertices embedded in an $L_p$ plane, the objective of the problem is to find a 2-connected network, spanning the given vertices and at most $k$ additional vertices, such that the length of the longest edge is minimised. In contrast to the discrete version of this problem the additional vertices may be located anywhere in the plane. The problem is motivated by the modelling of relay-augmentation for the optimisation of energy consumption in wireless ad hoc networks. Our algorithm employs Voronoi diagrams and properties of block-cut-vertex decompositions of graphs to find an optimal solution in $O(n^k\log^{\frac{5k}{2}}n)$ steps when $1<p<\infty$ and in $O(n^2\log^{\frac{7k}{2}+1}n)$ steps when $p\in\{1,\infty\}$.