Given a graph $G = (V, E)$, we wish to compute a spanning tree whose maximum vertex degree, i.e. tree degree, is as small as possible. Computing the exact optimal solution is known to be NP-hard, since it generalizes the Hamiltonian path problem. For the approximation version of this problem, a $\tilde{O}(mn)$ time algorithm that computes a spanning tree of degree at most $��^* +1$ is previously known [F��rer \& Raghavachari 1994]; here $��^*$ denotes the minimum tree degree of all the spanning trees. In this paper we give the first near-linear time approximation algorithm for this problem. Specifically speaking, we propose an $\tilde{O}(\frac{1}{��^7}m)$ time algorithm that computes a spanning tree with tree degree $(1+��)��^* + O(\frac{1}{��^2}\log n)$ for any constant $��\in (0,\frac{1}{6})$. Thus, when $��^*=��(\log n)$, we can achieve approximate solutions with constant approximate ratio arbitrarily close to 1 in near-linear time.

17 pages