A Cubic-Time 2-Approximation Algorithm for rSPR Distance
  Due to hybridization events in evolution, studying two different genes of a set of species may yield two related but different phylogenetic trees for the set of species. In this case, we want to measure the dissimilarity of the two trees. The rooted subtree prune and regraft (rSPR) distance of the two trees has been used for this purpose. The problem of computing the rSPR distance of two given trees has many applications but is unfortunately NP-hard. The previously best approximation algorithm for rSPR distance achieves a ratio of 2.5 and it was open whether a better approximation algorithm for rSPR distance exists. In this paper, we answer this question in the affirmative by presenting a cubic-time approximation algorithm for rSPR distance that achieves a ratio of 2. Our algorithm is based on the new notion of key and a number of new structural lemmas. The algorithm is fairly simple and the proof of its correctness is intuitively understandable albeit complicated. 
Zhi-Zhong Chen, Eita Machida, Lusheng Wang
cs.DS