An exact solution for finding minimum recombinant haplotype configurations on pedigrees with missing data by integer linear programming

We study the problem of reconstructing haplotype configurations from genotypes on pedigree data with missing alleles under the Mendelian law of inheritance and the minimum recombination principle, which is important for the construction of haplotype maps and genetic linkage/association analysis. Our previous results show that the problem of finding a minimum-recombinant haplotype configuration (MRHC) is in general NP-hard. The existing algorithms for MRHC either are heuristic in nature and cannot guarantee optimality, or only work under some restrictions (on e.g. the size and structure of the input pedigree, the number of marker loci, the number of recombinants in the pedigree, etc.). In addition, most of them cannot handle data with missing alleles and, for those that do consider missing data, they usually do not perform well in terms of minimizing the number of recombinants when a significant fraction of alleles are missing. In this paper, we develop an effective integer linear programming (ILP) formulation of the MRHC problem with missing data and a branch-and-bound strategy that utilizes a partial order relationship (and some other special relationships) among variables to decide the branching order. The partial order relationship is discovered in the preprocessing of constraints by considering unique properties in our ILP formulation. A directed graph is built based on the variables and their partial order relationship. By identifying and collapsing the strongly connected components in the graph, we may greatly reduce the size of an ILP instance. Non-trivial (lower and upper) bounds on the optimal number of recombinants are introduced at each branching node to effectively prune the search tree. When multiple solutions exist, a best haplotype configuration is selected based on a maximum likelihood approach. Our results on simulated data show that the algorithm could recover haplotypes with 50 loci from a pedigree of size 29 in seconds on a standard PC. Its accuracy is more than 99.8% for data with no missing alleles and 98.3% for data with 20% missing alleles in terms of correctly recovered phase information at each marker locus. As an application of our algorithm to real data, we present some test results on reconstructing haplotypes from a genome-scale SNP data set consisting of 12 pedigrees that have 0.8% to 14.5% missing alleles.