Polynomial-Time Algorithms for the Ordered Maximum Agreement Subtree Problem

Abstract

For a set of rooted, unordered, distinctly leaf-labeled trees, the NP-hard maximum agreement subtree problem (MAST) asks for a tree contained (up to isomorphism or homeomorphism) in all of the input trees with as many labeled leaves as possible. We study the ordered variants of MAST where the trees are uniformly or non-uniformly ordered. We provide the first known polynomial-time algorithms for the uniformly and non-uniformly ordered homeomorphic variants as well as the uniformly and non-uniformly ordered isomorphic variants of MAST. Our algorithms run in time $O(kn^3)$, $O(n^3 \min \{kn,\, n+\log^{k-1}n\})$, $O(kn^3)$, and $O(n^3 \min \{kn,\, n+\log^{k-1}n\})$, respectively, where n is the number of leaf labels and k is the number of input trees.