A Class Representative Model for Pure Parsimony Haplotyping

Abstract

H aplotyping 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 -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.

The pure parsimony haplotyping problem: overview and computational advances

“…In fact, exponential models imply the creation of too many variables and/or constraints for obtaining a solution within a reasonable time. On the other hand, apart from some exceptions (see Catanzaro et al, 2007), polynomial and hybrid models use quite weak lower bounds, so that closing the gap and terminating the branch-and-bound search is impractical within a reasonable time. We shall discuss exact approaches in Section 4.…”

“…The most recent polynomial model for PPH has been proposed by Catanzaro et al (2007) and is based on the class representatives with smallest index similar to the one proposed by Campelo et al (2008) for the node coloring problem (Garey and Johnson, 2003). The authors observed that D. Catanzaro and M. Labbe´/ Intl.…”

“…Specifically, the subsets are induced by the following four haplotypes: h 1 5 f1100g inducing the subset S 1 5 f1202g, h 2 5 f0011g inducing the subset S 2 5 f0212g, h 3 5 f1001g inducing the subset S 3 5 f2222, 1202g, and finally h 4 5 f0110g inducing the subset S 4 5 f2222, 0212g. Catanzaro et al (2007) exploited the bipartite graph representation of a solution to PPH to provide an ILP model that can be summarized as follows. Let associate an index with each subset S of genotypes induced by a haplotype h. Specifically, if i is the smallest index of a genotype belonging to S, then i is the index associated with S and the subset will be denoted as S i .…”

“…Finally, constraints (59)- (60) impose that the sum operation among haplotypes is respected for any pair of haplotypes h i and h j in the solution. Catanzaro et al (2007) discussed possible preprocessing techniques to reduce the overall number of variables and constraints, and provided valid inequalities to further strengthen the model. Computational experiments carried out on Brown and Harrower's benchmark instances (Brown and Harrower, 2006) evidenced that Formulation 7 outperforms Formulations 4 and 5, is characterized by a tight LP relaxation, and is able to handle instances containing hundreds of genotypes and SNPs.…”

A Preprocessing Procedure for Haplotype Inference by Pure Parsimony

“…Although some authors have noticed the need to introduce additional properties for the model [6] the HIPP still is a lively approach to HI as shown in the number of recently published papers [7], [8], [9]. In the literature we can find several solvers for the problem, particularly those based on integer linear programming (TIP [4], RTIP [4], PolyIP, [10], [11], HybridIP [10], SM, RM and SMM [7]), branch and bound (HAPAR [12]), genetic algorithms (GAHAP [13]), local search [14], boolean satisfiability, SAT (SHIPs [15]) and pseudo boolean optimization (PolyPB, RPoly and NRPoly [16], [17]). …”