On Improved Bounds for Bounded Degree Spanning Trees for Points in Arbitrary Dimension

Given points in Euclidean space of arbitrary dimension, we prove that there exists a spanning tree having no vertices of degree greater than 3 with weight at most 1.561 times the weight of the minimum spanning tree. We also prove that there is a set of points such that no spanning tree of maximal degree 3 exists that has this ratio be less than 1.447. Our central result is based on the proof of the following claim: Given n points in Euclidean space with one special point V , there exists a Hamiltonian path with an endpoint at V that is at most 1.561 times longer than the sum of the distances of the points to V. These proofs also lead to a way to find the tree in linear time given the minimal spanning tree. The minimum spanning tree (MST) problem in graphs is perhaps one of the most basic problems in graph algorithms. An MST is a spanning tree with minimal sum of edge weights. Efficient algorithms for finding an MST are well known. One variant on the MST problem is the bounded degree MST problem, which consists of finding a spanning tree satisfying given upper bounds on the degree of each vertex and with minimal sum of edges weights subject to these degree bounds. In general, this problem is NP-hard [1], so no efficient algorithm exists. However, there are certain achievable results. For undirected graphs, Singh and Lau [2] found a polynomial time algorithm to generate a spanning tree with total weight no more than that of the bounded degree MST and with each vertex having degree at most one greater than that vertex's bound. If the graph is undirected and satisfies the triangle inequality, Fekete and others [3] bound the ratio of the total weight of the bounded-degree MST to that of any