Rooted Maximum Agreement Supertrees

Given a set \(\mathcal{T}\) of rooted, unordered trees, where each \(T_{i} \in \mathcal{T}\) is distinctly leaf-labeled by a set Λ(T i ) and where the sets Λ(T i ) may overlap, the maximum agreement supertree problem (MASP) is to construct a distinctly leaf-labeled tree Q with leaf set \(\Lambda(Q)\subseteq \bigcup_{Ti\epsilon\mathcal{T}}\Lambda(Ti)\) such that |Λ(Q)| is maximized and for each \(T_{i}\in \mathcal{T}\), the topological restriction of T i to Λ(Q) is isomorphic to the topological restriction of Q to Λ(T i ). Let \(n = |\bigcup{T_{i}\in\mathcal{T}}\bigwedge(T_{i})|, k=|\mathcal{T}|, and D=maxt_{i}\in \mathcal{T}\{deg(T_{i}\}\). We first show that MASP with k = 2 can be solved in \(O(\sqrt{D}n {log}(2n/D))\) time, which is O(nlogn) when D = O(1) and O(n 1.5) 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.