Low-Degree Spanning Trees of Small Weight

TL;DR

This paper introduces new algorithms for constructing low-degree spanning trees with weights close to the minimum spanning tree, addressing a long-standing challenge in Euclidean graphs and improving approximation ratios.

Contribution

It provides the first algorithms with performance guarantees less than 2 for degree-3 and degree-4 spanning trees in Euclidean graphs, improving previous bounds.

Findings

Degree-3 spanning tree with at most 5/3 times MST weight.

Degree-4 spanning tree with at most 5/4 times MST weight.

Improved approximation ratios for Euclidean graphs.

Abstract

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.