Gilbert–Pollak conjecture

In mathematics, the Gilbert–Pollak conjecture is an unproven conjecture on the ratio of lengths of Steiner trees and Euclidean minimum spanning trees for the same point sets in the Euclidean plane.

It may be possible to construct a shorter network by using additional endpoints, not present in the given point set.

The Steiner ratio is the supremum, over all point sets, of the ratio of lengths of the Euclidean minimum spanning tree to the Steiner minimum tree.

Because the Steiner minimum tree is shorter, this ratio is always greater than one.

[2] A lower bound on the Steiner ratio is provided by three points at the vertices of an equilateral triangle of unit side length.

For these three points, the Euclidean minimum spanning tree uses two edges of the triangle, with total length two.

[2] The conjecture is famous for its proof by Ding-Zhu Du and Frank Kwang-Ming Hwang,[3][2] which later turned out to have a serious gap.

[4][5] Based on the flawed Du and Hwang result, J. Hyam Rubinstein and Jia F. Weng concluded that the Steiner ratio is also