On the Complexity of the Single Individual SNP Haplotyping Problem

We present several new results pertaining to haplotyping. These results concern the combinatorial problem of reconstructing haplotypes from incomplete and/or imperfectly sequenced haplotype fragments. We consider the complexity of the problems Minimum Error Correction (MEC) and Longest Haplotype Reconstruction (LHR) for different restrictions on the input data. Specifically, we look at the gapless case, where every row of the input corresponds to a gapless haplotype-fragment, and the 1gap case, where at most one gap per fragment is allowed. We prove that MEC is APX-hard in the 1-gap case and still NP-hard in the gapless case. In addition, we question earlier claims that MEC is NP-hard even when the input matrix is restricted to being completely binary. Concerning LHR, we show that this problem is NP-hard and APX-hard in the 1-gap case (and thus also in the general case), but is polynomial time solvable in the gapless case.