Long plane trees

Let $\mathcal{P}$ be a finite set of points in the plane in general position. For any spanning tree $T$ on $\mathcal{P}$, we denote by $|T|$ the Euclidean length of $T$. Let $T_{\text{OPT}}$ be a plane (that is, noncrossing) spanning tree of maximum length for $\mathcal{P}$. It is not known whether such a tree can be found in polynomial time. Past research has focused on designing polynomial time approximation algorithms, using low diameter trees. In this work we initiate a systematic study of the interplay between the approximation factor and the diameter of the candidate trees. Specifically, we show three results. First, we construct a plane tree $T_{\text{ALG}}$ with diameter at most four that satisfies $|T_{\text{ALG}}|\ge \delta\cdot |T_{\text{OPT}}|$ for $\delta>0.546$, thereby substantially improving the currently best known approximation factor. Second, we show that the longest plane tree among those with diameter at most three can be found in polynomial time. Third, for any $d\ge 3$ we provide upper bounds on the approximation factor achieved by a longest plane tree with diameter at most $d$ (compared to a longest general plane tree).