Vincent Berry

Given a collection of trees on n leaves with identical leaf set, the Mast, resp. Mct, problem consists in finding a largest subset of leaves such that all input trees restricted to these leaves are isomorphic, resp. have a common refinement. For Mast, resp. Mct, on k rooted trees, we give an O(min{3p kn,2.27p +kn 3}) exact algorithm, where p is the smallest number of leaves to remove from input trees in order for these trees to be isomorphic, resp. to admit a common refinement. This improves on [14] for Mast and proves fixed-parameter tractability for Mct. We also give an approximation algorithm for (the complement of) Mast similar to the one in [2], but with a better ratio and running time, and extend it to Mct. We generalize Mast and Mct to the case of supertrees where input trees can have non-identical leaf sets. For the resulting problems, Smast and Smct, we give an O(N+n) time algorithm for the special case of two input trees (N is the time bound for solving Mast, resp. Mct, on two O(n)-leaf trees). Last, we show that Smast and Smct parameterized in p are W[2]-hard and cannot be approximated in polynomial time within a constant factor unless P=NP, even when the input trees are rooted triples. We also extend the above results to the case of unrooted input trees.

Given a set of leaf-labelled trees with identical leaf sets, the well-known MAST problem consists of finding a subtree homeomorphically included in all input trees and with the largest number of leaves. MAST and its variant called MCT are of particular interest in computational biology. This paper presents positive and negative results on the approximation of MAST, MCT and their complement versions, denoted CMAST and CMCT. For CMAST and CMCT on rooted trees we give 3-approximation algorithms achieving significantly lower running times than those previously known. In particular, the algorithm for CMAST runs in linear time. The approximation threshold for CMAST, resp. CMCT, is shown to be the same whenever collections of rooted trees or of unrooted trees are considered. Moreover, hardness of approximation results are stated for CMAST, CMCT and MCT on small number of trees, and for MCT on unbounded number of trees.