Rooted maximum agreement supertrees

Given a set Τ of rooted, unordered trees, where each Ti ∈ Τ is distinctly leaf-labeled by a set Λ(Ti) and where the sets Λ(Ti) may overlap, the maximum agreement supertree problem (MASP) is to construct a distinctly leaf-labeled tree Q with leaf set Λ(Q) ⊆ ∪Ti ∈Τ Λ(T i) such that |Λ(Q)| is maximized and for each Ti ∈ Τ, the topological restriction of Ti to Λ(Q) is isomorphic to the topological restriction of Q to Λ(Ti). Let n = |∪Ti∈ΤΛ(Ti)|, k = |Τ|, and D = maxTi∈Τ{deg(Ti)}. We first show that MASP with k = 2 can be solved in O(√D n log(2n/D)) time, which is O(n log n) when D = O(1) and O(n1.5) when D is unrestricted. We then present an algorithm for MASP with D = 2 whose running time is polynomial if k = O(1). On the other hand, we prove that MASP is NP-hard for any fixed k ≥ 3 when D is unrestricted, and also NP-hard for any fixed D ≥ 2 when k is unrestricted even if each input tree is required to contain at most three leaves. Finally, we describe a polynomial-time (n/log n)-approximation algorithm for MASP.