Breakpoint Distance and PQ-Trees

Abstract

The PQ-tree is a fundamental data structure that can encode large sets of permutations. It has recently been used in comparative genomics to model ancestral genomes with some uncertainty: given a phylogeny for some species, extant genomes are represented by permutations on the leaves of the tree, and each internal node in the phylogenetic tree represents an extinct ancestral genome, represented by a PQ-tree. An open problem related to this approach is then to quantify the evolution between genomes represented by PQ-trees. In this paper we present results for two 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 permutations are given, with p ≥ 1, and we want to compute a permutation generated by the PQ-tree that minimizes the sum of the breakpoint distances to the p permutations. We show that this problem is Fixed-Parameter Tractable with respect to the breakpoint distance value. This last result applies both on signed and unsigned permutations, and to uni-chromosomal and multi-chromosomal permutations.