Approximation algorithms for the fixed-topology phylogenetic number problem

摘要

In the (ell)-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 (ell) connected components. The authors 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 (ell) (ge) 3 even for degree-3 trees in which no state labels more than (ell) + 1 leaves (and therefore there is a trivial (ell) + 1 phylogeny). They give a 2-approximation algorithm for all (ell) (ge) 3 for arbitrary input topologies and they given 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 (ell)-phylogeny problems. The 2-approximation algorithm is the first application of linear programming to approximation algorithms for phylogeny problems. They extend their results to a related problem in which characters are polymorphic.