Breakpoint distance and PQ-trees

Abstract

The PQ-tree is a fundamental data structure that has also been used in comparative genomics to model ancestral genomes with some uncertainty. To quantify the evolution between genomes represented by PQ-trees, in this paper we study two fundamental problems of PQ-tree comparison motivated by this application. First, we show that the problem of comparing two PQ-trees by computing the minimum breakpoint distance among all pairs of permutations generated respectively by the two considered PQ-trees is NP-complete for unsigned permutations. Next, we consider a generalization of the classical Breakpoint Median problem, where an ancestral genome is represented by a PQ-tree and p≥1 permutations are given and we want to compute a permutation generated by the PQ-tree that minimizes the sum of the breakpoint distances to the p permutations (or k). We show that this problem is also NP-complete for p≥2, and is fixed-parameter tractable with respect to k for p≥1.