Fixed Parameter Polynomial Time Algorithms for Maximum Agreement and   Compatible Supertrees
  Consider a set of labels $L$ and a set of trees ${\mathcal T} = \{{\mathcal T}^{(1), {\mathcal T}^{(2), ..., {\mathcal T}^{(k) \$ where each tree ${\mathcal T}^{(i)$ is distinctly leaf-labeled by some subset of $L$. One fundamental problem is to find the biggest tree (denoted as supertree) to represent $\mathcal T}$ which minimizes the disagreements with the trees in ${\mathcal T}$ under certain criteria. This problem finds applications in phylogenetics, database, and data mining. 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 the first polynomial time algorithms for both MASP and MCSP when both $k$ and the maximum degree $D$ of the trees are constant. 
Viet Tung Hoang, Wing-Kin Sung