Generalised k-Steiner Tree Problems in Normed Planes

TL;DR

This paper generalizes the 1-Steiner tree problem to include up to k Steiner points in normed planes, providing algorithms with polynomial time complexity for solving these extended problems.

Contribution

It extends existing methods to solve k-Steiner tree problems in various normed planes, broadening the scope of Steiner tree solutions.

Findings

Solutions for k-Steiner tree problems in polynomial time for fixed k

Extension of methods to normed planes beyond Euclidean

Applicable to k-bottleneck Steiner tree and other variants

Abstract

The 1-Steiner tree problem, the problem of constructing a Steiner minimum tree containing at most one Steiner point, has been solved in the Euclidean plane by Georgakopoulos and Papadimitriou using plane subdivisions called oriented Dirichlet cell partitions. Their algorithm produces an optimal solution within $O(n^2)$ time. In this paper we generalise their approach in order to solve the $k$-Steiner tree problem, in which the Steiner minimum tree may contain up to $k$ Steiner points for a given constant $k$. We also extend their approach further to encompass other normed planes, and to solve a much wider class of problems, including the $k$-bottleneck Steiner tree problem and other generalised $k$-Steiner tree problems. We show that, for any fixed $k$, such problems can be solved in $O(n^{2k})$ time.