Near-linear Time Algorithm for Approximate Minimum Degree Spanning Trees

TL;DR

This paper presents the first near-linear time algorithm for approximating the minimum degree spanning tree problem, achieving near-optimal solutions efficiently for graphs with large minimum degrees.

Contribution

It introduces a near-linear time approximation algorithm that produces spanning trees with degrees close to the minimum, improving efficiency over previous methods.

Findings

Achieves near-linear time complexity of (m) for the approximation algorithm.

Provides solutions with degree at most (1+psilon)elta* + O(rac{1}{psilon^2}log n).

Works effectively when elta*=ig(ig(psilonig)ig).

Abstract

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