The Gilbert and Pollak conjecture—a generalization

Gilbert and Pollak conjectured that the length of the minimal Steiner tree (MST) is at least \documentclass{article}\pagestyle{empty}\begin{document}$ \sqrt {3/4} $\end{document} that of the minimal spanning tree. In a subsequent paper, Gilbert generalized the Steiner tree problem by adding flow dependent weights to the arcs. We refer to the generalized problem as the Minimal Gilbert Network Problem, and to its optimal solution as the Minimal Gilbert Network (MGN). The purpose of this article is to generalize Gilbert and Pollak's conjecture to the ratio between the MGN and the regular minimal network (where extra nodes are not allowed). We prove that when the regular minimal network connects three nodes, the highest improvement possible, by adding exactly one extra point, is (2 − \documentclass{article}\pagestyle{empty}\begin{document}$ \sqrt 3 $\end{document})/2, and that this maximal improvement can be achieved only for the symmetric case, namely, where the three nodes are the vertices of an equilateral triangle and the weights of the three arcs are equal. As this is actually a Steiner tree case, we believe that the generalized and the original conjectures are equivalent, i.e., if this generalized conjecture will ever be disproved, the counterexample (or at least one of the possible counterexamples) would also disprove the original conjecture. Another result we present is a proof of the existence of the Gilbert generalized Steiner construction.