In the ℓ-phylogeny problem, one wishes to construct an evolutionary tree for a set of species represented by characters, in which each state of each character induces no more than ℓ connected components. We consider the fixed-topology version of this problem for fixed-topologies of arbitrary degree. This version of the problem is known to be NP-complete for ℓ ≥ 3 even for degree-3 trees in which no state labels more than ℓ+1 leaves (and therefore there is a trivial ℓ+1 phylogeny). We give a 2-approximation algorithm for all ℓ ≥ 3 for arbitrary input topologies and we give an optimal approximation algorithm that constructs a 4-phylogeny when a 3-phylogeny exists. Dynamic programming techniques, which are typically used in fixed-toplogy problems, cannot be applied to ℓ-phylogeny problems. Our 2-approximation algorithm is the first application of linear programming to approximation algorithms for phylogeny problems. We extend our results to a related problem in which characters are polymorphic.
CITATION STYLE
Cryan, M., Goldberg, L. A., & Phillips, C. A. (1997). Approximation algorithms for the fixed-topology phylogenetic number problem. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 1264, pp. 130–149). Springer Verlag. https://doi.org/10.1007/3-540-63220-4_56
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