Mary Cryan, Leslie Ann Goldberg and Cynthia A. Phillips, Approximation Algorithms for the Fixed-Topology Phylogenetic Number Problem (June 1, 1997).
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 when L is at least 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 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-topology 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.
<%@ include file="cited.html" %>M. Cryan, L.A. Goldberg and C.A. Phillips, "Approximation Algorithms for the Fixed-Topology Phylogenetic Number Problem", Proceedings of Combinatorial Pattern Matching 8, (1997)
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