International Journal of Foundations of Computer Science c  World Scientiﬁc Publishing Company Approximating the Maximum Isomorphic Agreement Subtree is Hard Paola Bonizzoni Dipartimento di Informatica, Sistemistica e Comunicazione, Universit`a degli Studi di Milano - Bicocca, via Bicocca degli Arcimboldi 8, 20126 Milano(Italy), bonizzoni@disco.unimib.it Gianluca Della Vedova Dipartimento di Informatica, Sistemistica e Comunicazione, Universit`a degli Studi di Milano - Bicocca, via Bicocca degli Arcimboldi 8, 20126 Milano(Italy), dellavedova@disco.unimib.it and Giancarlo Mauri Dipartimento di Informatica, Sistemistica e Comunicazione, Universit`a degli Studi di Milano - Bicocca, via Bicocca degli Arcimboldi 8, 20126 Milano(Italy), mauri@disco.unimib.it Received Revised Communicated by ABSTRACT The Maximum Isomorphic Agreement Subtree (MIT) problem is one of the simplest versions of the Maximum Interval Weight Agreement Subtree method (MIWT) which is used to compare phylogenies. More precisely MIT allows to provide a subset of the species such that the exact distances between species in such subset are preserved among all evolutionary trees considered. In this paper, the approximation complexity of the MIT problem is investigated, showing that it cannot be approximated in polynomial time within factor log δ n for any δ> 0 unless NP⊆DTIME(2 polylog n ) for instances containing three trees. Moreover, we show that such result can be strengthened whenever instances of the MIT problem can contain an arbitrary number of trees, since MIT shares the same approximation lower bound of MAX CLIQUE. Keywords: computational complexity, bioinformatics, inapproximability, evolutionary tree comparison. 1. Introduction Evolutionary trees are trees where each leaf is labeled by a distinct element in a set S of species and where all internal nodes have degree at least three. They are frequently used by biologists to represent classiﬁcations of species. More pre- cisely, each edge is weighted with the estimated (temporal) distance between the two species represented by its endpoints. A number of methods to infer evolution- ary trees have been proposed [9, 15, 5, 13, 14, 17, 31, 4, 3, 27, 8, 34], moreover it is rather common to compare the same set of species w.r.t. diﬀerent biological 1