Approximation algorithms for the fixed-topology phylogenetic number problem

In the l-phylogeny problem, one wishes to construct an evolutionary tree fo r a set of species represented by characters, in which each state of each c haracter 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 a l greater tha n or equal to 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 greater than or equal to 3 for arbitrary input topologies and we give an optimal approximation algorithm that constr ucts a 4-phylogeny when a 3-phylogeny exists. Dynamic programming technique s, which are typically used in fixed-topology problems, cannot be applied t o l-phylogeny problems. Our 2-approximation algorithm is the first applicat ion of linear programming to approximation algorithms for phylogeny problem s. We extend our results to a related problem in which characters are polym orphic.