Parameterized and approximation algorithms for maximum agreement forest in multifurcating trees

We study parameterized algorithms and approximation algorithms for the maximum agreement forest problem, which, for two given leaf-labeled trees, is to find a maximum forest that is a subgraph of both trees. The problem was motivated by research in phylogenetics. For parameterized algorithms, while the problem is known to be fixed-parameter tractable for binary trees, it was an open problem whether the problem is still fixed-parameter tractable for general trees. We resolve this open problem by developing an O(3kn)O(3kn)-time parameterized algorithm for general trees. Our techniques on tree structures also lead to a polynomial-time approximation algorithm of ratio 3 for the problem, giving the first constant-ratio approximation algorithm for general trees.