Hardness of Covering Alignment: Phase Transition in Post-Sequence Genomics

Covering alignment problems arise from recent developments in genomics; so called pan-genome graphs are replacing reference genomes, and advances in haplotyping enable full content of diploid genomes to be used as basis of sequence analysis. In this paper, we show that the computational complexity will change for natural extensions of alignments to pan-genome representations and to diploid genomes. More broadly, our approach can also be seen as a minimal extension of sequence alignment to labelled directed acyclic graphs (labeled DAGs). Namely, we show that finding a <italic>covering alignment</italic> of two labeled DAGs is NP-hard even on binary alphabets. A covering alignment asks for two paths <inline-formula><tex-math notation="LaTeX">$R_1$</tex-math><alternatives><inline-graphic xlink:href="makinen-ieq1-2831691.gif"/></alternatives></inline-formula> (red) and <inline-formula><tex-math notation="LaTeX">$G_1$</tex-math><alternatives><inline-graphic xlink:href="makinen-ieq2-2831691.gif"/></alternatives></inline-formula> (green) in DAG <inline-formula><tex-math notation="LaTeX">$D_1$</tex-math><alternatives><inline-graphic xlink:href="makinen-ieq3-2831691.gif"/></alternatives></inline-formula> and two paths <inline-formula><tex-math notation="LaTeX">$R_2$</tex-math><alternatives><inline-graphic xlink:href="makinen-ieq4-2831691.gif"/></alternatives></inline-formula> (red) and <inline-formula><tex-math notation="LaTeX">$G_2$</tex-math><alternatives><inline-graphic xlink:href="makinen-ieq5-2831691.gif"/></alternatives></inline-formula> (green) in DAG <inline-formula><tex-math notation="LaTeX">$D_2$</tex-math><alternatives><inline-graphic xlink:href="makinen-ieq6-2831691.gif"/></alternatives></inline-formula> that cover the nodes of the graphs and maximize the sum of the global alignment scores: <inline-formula><tex-math notation="LaTeX">$\mathsf {as}(\mathsf {sp}(R_1),\mathsf {sp}(R_2))+\mathsf {as}(\mathsf {sp}(G_1),\mathsf {sp}(G_2))$</tex-math><alternatives><inline-graphic xlink:href="makinen-ieq7-2831691.gif"/></alternatives></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$\mathsf {sp}(P)$</tex-math><alternatives><inline-graphic xlink:href="makinen-ieq8-2831691.gif"/></alternatives></inline-formula> is the concatenation of labels on the path <inline-formula><tex-math notation="LaTeX">$P$</tex-math><alternatives><inline-graphic xlink:href="makinen-ieq9-2831691.gif"/></alternatives></inline-formula>. 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 recombinations. We also give a reduction to the other direction, to show that such a recombination-oblivious diploid alignment is NP-hard on alphabets of size 3.

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