Near-linear Time Algorithm for Approximate Minimum Degree Spanning Trees

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 ${\mathrm{\Delta }}^{\ast }+1$ is previously known [F"urer & Raghavachari 1994]; here ${\mathrm{\Delta }}^{\ast }$ 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 $\tilde{O}(\frac{1}{{ϵ}^7}m)$ time algorithm that computes a spanning tree with tree degree $(1+\epsilon)\Delta^* + O(\frac{1}{\epsilon^2}\log n)$foranyconstant$\epsilon \in (0,\frac{1}{6})$. Thus, when ${\mathrm{\Delta }}^{\ast }=\omega (\log n)$, we can achieve approximate solutions with constant approximate ratio arbitrarily close to 1 in near-linear time.