Near-linear Time Algorithms for Approximate Minimum Degree Spanning Trees

12/26/2017

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by   Ran Duan, et al.

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Given a graph G = (V, E), n=|V|, m=|E|, we wish to compute a spanning tree whose maximum vertex 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 Õ(mn) time algorithm that computes a spanning tree of degree at most Δ^* +1 is previously known [Fürer, Raghavachari 1994]; here Δ^* denotes the optimal tree degree. In this paper we give the first near-linear time algorithm for this problem. Specifically speaking, we first propose a simple Õ(m) time algorithm that achieves an O(Δ^* n) approximation; then we further improve this algorithm to obtain a (1+δ)Δ^* + O(1/δ^2 n) approximation in Õ(1/δ^6m) time.