20.500.12556/RUL-177721 Long plane trees In the longest plane spanning tree problem, we are given a finite planar point set ${\mathcal P}$, and our task is to find a plane (i.e., noncrossing) spanning tree for ${\mathcal P}$ with maximum total Euclidean edge length. Despite more than two decades of research, it remains open whether this problem is NP-hard. Thus, previous results have focused on polynomial-time algorithms that produce plane trees whose total edge length approximates $\text{OPT}$, the maximum possible length. The approximate trees in these algorithms all have small unweighted diameter, typically two to four. It is natural to ask whether this is a common feature of longest plane spanning trees, or an artifact of the specific approximation algorithms. We provide three results to elucidate the interplay between the approximation guarantee and the unweighted diameter of the approximate trees. First, we describe a polynomial-time algorithm to construct a plane tree with diameter at most four and total edge length at least $0.546\cdot\text{OPT}$. This constitutes a substantial improvement over the state of the art. Second, we show that a longest plane tree among those with diameter at most three can be found in polynomial time. Third, for any candidate diameter $d \ge 3$, we provide upper bounds on the approximation factor that can be achieved by a longest plane tree with diameter at most $d$ (compared to a longest plane tree without constraints). computational geometry geometric network design spanning trees plane straight-line graphs approximation algorithms true false true Angleški jezik Angleški jezik Članek v reviji 2026-01-05 11:20:51 2026-01-05 11:20:56 2026-01-06 04:17:12 0000-00-00 00:00:00 2026 0 0 Str. 5:1-5:40 iss. 1, art. 5 Vol. 22 Jan. 2026 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2026-01-01 519.17:004 1549-6325 10.1145/3765740 263386883 3765740.pdf 3765740.pdf 1 8B22EB559477C11B3A98F9638478C08F c072f79f6bbf124b327850d3d1db8ce2d8801606615486b44b1abe9cb87c2a5a 1b624e4b-ea29-11f0-bf92-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=224956 https://dl.acm.org/doi/10.1145/3765740 1 3a369f7d-ea20-11f0-bf92-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=224952 Fakulteta za matematiko in fiziko 0 0 0