The degree-d spanning tree problem asks for a minimum-weight spanning tree in
which the degree of each vertex is at most d. When d=2 the problem is TSP, and
in this case, the well-known Christofides algorithm provides a
1.5-approximation algorithm (assuming the edge weights satisfy the triangle
inequality).
  In 1984, Christos Papadimitriou and Umesh Vazirani posed the challenge of
finding an algorithm with performance guarantee less than 2 for Euclidean
graphs (points in R^n) and d > 2. This paper gives the first answer to that
challenge, presenting an algorithm to compute a degree-3 spanning tree of cost
at most 5/3 times the MST. For points in the plane, the ratio improves to 3/2
and the algorithm can also find a degree-4 spanning tree of cost at most 5/4
times the MST.Comment: conference version in Symposium on Theory of Computing (1994

Balaji Raghavachari

Garey Michael R.

Neal Young

Samir Khuller

English

arXiv

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 $K \u3e 2$. 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 $d \geqslant 3$, 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

Khuller, Samir

Raghavachari, Balaji

Young, Neal

Dartmouth Digital Commons (Dartmouth College)

Low-Degree Spanning Trees of Small Weight

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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 &gt; 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 ≥ 3, an arbitrary collection of points in ℛd 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

eScholarship - University of California

Low-Degree Spanning Trees of Small Weight

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Dartmouth Digital Commons (Dartmouth College)

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eScholarship - University of California