Aligning and Labeling Genomes under the Duplication-Loss Model

Abstract

We investigate the complexity of two combinatorial problems related to pairwise genome alignment under the duplication-loss model of evolution. Under this model, the unaligned parts of the genomes (gaps) are interpreted as duplications and losses. The first, and most general, combinatorial problem that we consider is the Duplication-Loss Alignment problem, which is to find an alignment of minimum duplication-loss cost. Defining the cost as the number of segmental duplications and individual losses, the problem has been recently shown NP-hard. Here, we improve this result by showing that the Duplication-Loss Alignment (DLA) problem is APX-hard even if the number of occurrences of a gene inside a genome is bounded by 2. We then consider a more constrained version, the Feasible Relabeling Alignment problem, involved in a general methodology for solving DLA, that aims to infer a feasible (in term of evolutionary history) most parsimonious relabeling of an initial best candidate labeled alignment which is potentially cyclic. We show that it is equivalent to Minimum Feedback Vertex Set on Directed Graph, hence implying that the problem is APX-hard, is fixed-parameter tractable and approximable within factor O(log|χ|loglog|χ|), where χ is the aligned genome considered by Feasible Relabeling Alignment.