The Matroid Structure of Representative Triple Sets and Triple-Closure Computation

The closure $\textrm{cl}(R)$ of a consistent set $R$ of triples (rooted binary trees on three leaves) provides essential information about tree-like relations that are shown by any supertree that displays all triples in $R$. In this contribution, we are concerned with representative triple sets, that is, subsets $R'$ of $R$ with $\textrm{cl}(R') = \textrm{cl}(R)$. In this case, $R'$ still contains all information on the tree structure implied by $R$, although $R'$ might be significantly smaller. We show that representative triple sets that are minimal w.r.t.\ inclusion form the basis of a matroid. This in turn implies that minimal representative triple sets also have minimum cardinality. In particular, the matroid structure can be used to show that minimum representative triple sets can be computed in polynomial time with a simple greedy approach. For a given triple set $R$ that "identifies" a tree, we provide an exact value for the cardinality of its minimum representative triple sets. In addition, we utilize the latter results to provide a novel and efficient method to compute the closure $\textrm{cl}(R)$ of a consistent triple set $R$ that improves the time complexity $\mathcal{O}(|R||L_R|^4)$ of the currently fastest known method proposed by Bryant and Steel (1995). In particular, if a minimum representative triple set for $R$ is given, it can be shown that the time complexity to compute $\textrm{cl}(R)$ can be improved by a factor up to $|R||L_R|$. As it turns out, collections of quartets (unrooted binary trees on four leaves) do not provide a matroid structure, in general.