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Improved algorithms for maximum agreement and compatible supertrees. (English)
Algorithmica 59, No. 2, 195-214 (2011).
Summary: Consider a set of labels $L$ and a set of unordered trees ${\cal T}=\{{\cal T}^{(1)},{\cal T}^{(2)},\dots ,{\cal T}^{(k)}\}$ where each tree ${\cal T}^{(i)}$ is distinctly leaf-labeled by some subset of $L$. One fundamental problem is to find the biggest tree (denoted as supertree) to represent ${\cal T}$ which minimizes the disagreements with the trees in ${\cal T}$ under certain criteria. In this paper, we focus on two particular supertree problems, namely, the Maximum Agreement Supertree Problem (MASP) and the Maximum Compatible Supertree Problem (MCSP). These two problems are known to be NP-hard for $k\geq 3$. This paper gives improved algorithms for both MASP and MCSP. In particular, our results imply the first polynomial time algorithms for both MASP and MCSP when both $k$ and the maximum degree $D$ of the input trees are constant.