A Sublinear Approximation Algorithm for Minimum Dilation Trees in the Plane
TL;DR
The first sublinear approximation algorithm for the minimum dilation tree in the Euclidean plane is given, whose approximation ratio is $\tilde{O}(n^{14/15})$ and the algorithm runs in polynomial time.
Abstract
The dilation of a geometric graph measures how much longer the path between pairs of points becomes when restricted to graph edges, rather than following the direct path through the ambient space. The minimum dilation tree of a point set is the spanning tree with minimum dilation, where edge lengths in the tree are given by distances in the ambient space. In the Euclidean plane, computing the minimum dilation tree is NP-hard, but no hardness of approximation result is known. On the other hand, the minimum spanning tree is an $(n-1)$-approximation to the minimum dilation tree, but no asymptotically-better approximation algorithm is known for general point sets in the Euclidean plane. We give the first sublinear approximation algorithm for the minimum dilation tree in the Euclidean plane. Our approximation ratio is $\tilde{O}(n^{14/15})$ and our algorithm runs in polynomial time. This resolves an open problem proposed by Eppstein in 1996.