Complexity and approximation of the minimum recombinant haplotype configuration problem

We study the complexity and approximation of the problem of reconstructing haplotypes from genotypes on pedigrees under the Mendelian Law of Inheritance and the minimum recombinant principle (MRHC). First, we show that the MRHC for simple pedigrees where each member has at most one mate and at most one child (i.e. binary-tree pedigrees) is NP-hard. Second, we present some approximation results for the MRHC problem, which are the first approximation results in the literature to the best of our knowledge. We prove that the MRHC on two-locus pedigrees or binary-tree pedigrees with missing data cannot be approximated unless P=NP. Next we show that the MRHC on two-locus pedigrees without missing data cannot be approximated within any constant ratio under the Unique Games Conjecture and can be approximated within the ratio O(log(n)). Our L-reduction for the approximation hardness gives a simple alternative proof that the MRHC on two-locus pedigrees is NP-hard, which is much easier to understand than the original proof. We also show that the MRHC for tree pedigrees without missing data cannot be approximated within any constant ratio under the Unique Games Conjecture, too. Finally, we explore the hardness and approximation of the MRHC on pedigrees where each member has a bounded number of children and mates mirroring real pedigrees.