On the Generalised Character Compatibility Problem for Non-branching Character Trees

Abstract

In [13], the authors introduced the Generalised Character Compatibility Problem as a generalisation of the Perfect Phylogeny Problem for a set of species. This generalised problem takes into account the fact that while a species may not be expressing a certain trait, i.e., having teeth, its DNA may contain data for this trait in a non-functional region. The authors showed that the Generalised Character Compatibility Problem is NP-complete for an instance of the problem involving five states, where the characters’ state transition trees are branching. They also presented a class of instances of the problem that is polynomial-time solvable. The authors posed an open problem about the complexity of this problem when no branching is allowed in the character trees. They answered this question in [2], where they showed that for an instance in which each character tree is 0→1→2 (no branching), and only the states {1},{0,2},{0,1,2} are allowed, is NP-complete. This, however, does not provide an answer to the exact question posed in [3], which allows only one type of generalised state: {0,2}, called here the Benham-Kannan-Warnow (BKW) Case.

In this paper, we study the complexity of various versions of this problem with non-branching character trees, depending on the set of states allowed, and depending on the restriction on the phylogeny tree: any tree, path or single-branch tree. In particular, we show that if the phylogeny tree is required to have only one branch: (a) the problem still remains NP-complete (for instance with states {1},{0,2},{0,1,2}), and (b) the problem is polynomial-time solvable in the BKW Case (with states {0},{1},{2},{0,2}). We show the second result by unveiling a surprising connection to the Consecutive-Ones Property (C1P) Problem, used for instance, in DNA physical mapping, interval graph recognition and data retrieval.