Low-Degree Spanning Trees of Small Weight

Given $n$ points in the plane, the degree-$K$ spanning-tree problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most $K$. This paper addresses the problem of computing low-weight degree-$K$ spanning trees for $K>2$. It is shown that for an arbitrary collection of $n$ points in the plane, there exists a spanning tree of degree 3 whose weight is at most 1.5 times the weight of a minimum spanning tree. It is shown that there exists a spanning tree of degree 4 whose weight is at most 1.25 times the weight of a minimum spanning tree. These results solve open problems posed by Papadimitriou and Vazirani. Moreover, if a minimum spanning tree is given as part of the input, the trees can be computed in $O(n)$ time. The results are generalized to points in higher dimensions. It is shown that for any $d \ge 3$, an arbitrary collection of points in $\Re^d$ contains a spanning tree of degree 3 whose weight is at most 5/3 times the weight of a minimum spanning tree. This is the first paper that achieves factors better than 2 for these problems.