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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 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 MRHC on two-locus pedigrees or binary-tree pedigrees with missing data cannot be approximated (the formal definition is given in section 1.2) unless P=NP. Next we show that 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 ratio O$(\sqrt{{\rm log}(n)})$. Our L-reduction for the approximation hardness gives a simple alternative proof that MRHC on two-locus pedigrees is NP-hard, which is much easier to understand than the original proof. We also show that 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 MRHC on pedigrees where each member has a bounded number of children and mates mirroring real pedigrees.

Complexity and Approximation of the Minimum Recombination Haplotype Configuration Problem