, Volume 68, Issue 4, pp 886-915
Date: 08 Nov 2012

Constructing Minimal Phylogenetic Networks from Softwired Clusters is Fixed Parameter Tractable

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Here we show that, given a set of clusters \({\mathcal{C}}\) on a set of taxa \({\mathcal{X}}\) , where \(|{\mathcal{X}}|=n\) , 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 \({\mathcal{C}}\) in the softwired sense, and if so to construct such a network. This extends a result from Kelk et al. (in IEEE/ACM Trans. Comput. Biol. Bioinform. 9:517–534, 2012) which showed that the problem is polynomial-time solvable for fixed k. By defining “k-reticulation generators” analogous to “level-k generators”, 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.