Diploid Alignment is NP-hard

Human genomes consist of pairs of homologous chromosomes, one of which is inherited from the mother and the other from the father. Species, such as human, with pairs of homologous chromosomes are called diploid. Sequence analysis literature is, however, almost solely built under the model of a single haplotype sequence representing a species. This fundamental choice is apparently due to the huge conceptual simplification of carrying out analyses over sequences rather than over pairs of related sequences. In this paper, we show that not only raising the abstraction level creates conceptual difficulties, but also the computational complexity will change for a natural non-trivial extension of optimal alignment to diploids. As of independent interest, our approach can also be seen as an extension of sequence alignment to labelled directed acyclic graphs (labeled DAGs). Namely, we show that a covering alignment of two labeled DAGs is NP-hard. A covering alignment is to find two paths P1(A) and P2(A) in DAG A and two paths P1(B) and P2(B) in DAG B that cover the nodes of the graphs and maximize sum of the global alignment scores: S(`(P1(A)), `(P1(B))) + S(`(P2(A)), `(P2(B))), where `(P ) is the concatenation of labels on the path P . Pair-wise alignment of haplotype sequences forming a diploid chromosome can be converted to a two-path coverable labelled DAG, and then the covering alignment models the similarity of two diploids over arbitrary recombination. ar X iv :1 61 1. 05 08 6v 1 [ cs .C C ] 1 5 N ov 2 01 6