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 $p\in{1,\infty}$.

💡 Research Summary

The paper addresses the “relay‑augmentation” problem that arises in wireless ad‑hoc networks, where a limited number of additional relay nodes (Steiner points) may be placed anywhere in the plane to reduce the longest transmission distance (the bottleneck) while guaranteeing a prescribed level of survivability. Specifically, the authors study the geometric bottleneck 2‑connected k‑Steiner network problem: given a set X of n terminals embedded in an Lₚ plane (1 ≤ p ≤ ∞) and an integer k > 2 (treated as a constant), construct a 2‑connected network spanning X together with at most k additional Steiner points such that the length of the longest edge, measured in the Lₚ metric, is minimized.

Prior work had only solved the analogous 1‑connected version (single‑path survivability) in polynomial time when k is constant, while the 2‑connected case remained open. The authors fill this gap by presenting the first exact polynomial‑time algorithm, called the 2‑Bottleneck algorithm, and by proving its correctness and time bounds.

Main ideas

1. Underlying networks – The algorithm first discards all Steiner edges and works with a pure terminal graph, called an underlying network. All possible underlying networks are generated by considering, for each distance ℓᵢ in the sorted list of pairwise terminal distances, the subgraph