Approximating Spanning Trees with Few Branches

Given an undirected, connected graph, the aim of the minimum branch-node spanning tree problem is to find a spanning tree with the minimum number of nodes of degree larger than 2. The problem is motivated by network design problems where junctions are significantly more expensive than simple end- or through-nodes, and are thus to be avoided. Unfortunately, it is NP-hard to recognize instances that admit an objective value of zero, rendering the search for guaranteed approximation ratios futile.We suggest to investigate a complementary formulation, called maximum path-node spanning tree, where the goal is to find a spanning tree that maximizes the number of nodes with degree at most two. While the optimal solutions (and the practical applications) of both formulations coincide, our formulation proves more suitable for approximation. In fact, it admits a trivial 1/2-approximation algorithm. Our main contribution is a local search algorithm that guarantees a ratio of 6/11, as well as showing that the problem is APX-hard, i.e., it does not allow a PTAS.