Constructing minimal phylogenetic networks from softwired clusters is fixed parameter tractable

Here we show that, given a set of clusters C on a set of taxa X, where |X|=n, it is possible to determine in time f(k).poly(n) whether there exists a level-<= k network (i.e. a network where each biconnected component has reticulation number at most k) that represents all the clusters in C in the softwired sense, and if so to construct such a network. This extends a polynomial time result from "On the elusiveness of clusters" by Kelk, Scornavacca and Van Iersel(2011). By generalizing the concept of "level-k generator" to general networks, we then extend this fixed parameter tractability result to the problem where k refers not to the level but to the reticulation number of the whole network.