Fixed Parameter Polynomial Time Algorithms for Maximum Agreement and Compatible Supertrees

TL;DR

This paper introduces fixed parameter polynomial time algorithms for the maximum agreement and compatible supertree problems, which are NP-hard in general, when the number of trees and their maximum degree are fixed.

Contribution

It provides the first polynomial time algorithms for MASP and MCSP with fixed number of trees and bounded degree, advancing computational methods in phylogenetics and data analysis.

Findings

Algorithms run in polynomial time for fixed k and D.

First polynomial algorithms for these supertree problems.

Applicable to phylogenetics, databases, and data mining.

Abstract

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.