Minimum Recombiant Haplotype Configuration on Tree Pedigrees

We study the problem of reconstructing haplotype configurations from genotypes on pedigree data under the Mendelian law of inheritance and the minimum recombination principle, which is very important for the construction of haplotype maps and genetic linkage/association analysis. Li and Jiang [9,10] recently proved that the Minimum Recombinant Haplotype Configuration (MRHC) problem is NP-hard, even if the number of marker loci is 2. However, the proof uses pedigrees that contain complex mating loop structures that are not common in practice. The complexity of MRHC in the loopless case was left as an open problem. In this paper, we show that loopless MRHC is NP-hard. We also present two dynamic programming algorithms that can be useful for solving loopless MRHC (and general MRHC) in practice. The first algorithm performs dynamic programming on the members of the input pedigree and is efficient when the number of marker loci is bounded by a small constant. It takes advantage of the tree structure in a loopless pedigree. The second algorithm performs dynamic programming on the marker loci and is efficient when the number of the members of the input pedigree is small. This algorithm also works for the general MRHC problem. We have implemented both algorithms and applied the first one to both simulated and real data. Our preliminary experiments demonstrate that the algorithm is often able to solve MRHC efficiently in practice.