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Similarity and diversity among individuals of the same species are expressed in small DNA variations called Single Nucleotide Polymorphism. The knowledge of SNP phase gives rise to the haplotyping problem that in the parsimonious version states to infer the minimum number of haplotypes from a given set of genotype data. ILP technique represents a good resolution strategy for this interesting combinatorial problem whose main limit lies in its NP-hardness. In this paper we present a new polynomial model for the haplotyping inference by parsimony problem characterized by the original use of a maximum formulation jointly with a good heuristic solution. This approach showed to be a robust basic model that can be used as starting point for more sophisticated ILP techniques like branch and cut and polyhedral studies.

Haplotyping estimation from aligned single nucleotide polymorphism fragments has attracted increasing attention in recent years because of its importance in the analysis of fine-scale genetic data. Its application fields range from mapping of complex disease genes to inferring population histories, passing through designing drugs, functional genomics, and pharmacogenetics. The literature proposes several criteria for haplotyping populations, each of them characterized by biological motivations. One of the most important haplotyping criteria is parsimony, which consists of finding the minimum number of haplotypes necessary to explain a given set of genotypes. Parsimonious haplotype estimation is an 𝒩𝒫-hard problem for which the literature has proposed several integer programming (IP) models. Here, we describe a new polynomial-sized IP model based on the concept of class representatives, already used for the coloring problem. We propose valid inequalities to strengthen our model and show, through computational experiments, that our model outperforms the best IP models currently known in literature.