The genome of every higher developed species, whether animal, plant or fungus, is based on a species-dependent number of homologous sets of chromosomes. The number of sets ranges from at least one set as it is found in yeast, which is a haploid species, followed by two sets in human (diploid) to much bigger numbers like four sets (tetraploid) in some varieties of the potato (e.g. Solanum tuberosum) or six homologous sets (hexaploid) in wheat (e.g. Triticum aestivum). The strawberry (e.g. Fragaria ananassa) can even have eight sets (octaploid). Chromosomes are sequences over the nucleotide alphabet, where the position of a specific nucleotide on the chromosome is called site or locus (Figure 1).

Single Nucleotide Polymorphism or SNP is a DNA sequence variation, occurring when a single nucleotide is altered 7. Thus, a site in a population of a species is a SNP site if at least a second sort of nucleotide occurs at this site at least once.

An allele is a different form of some segment of a chromosome, such as a second sort of nucleotide at a SNP site. Here, we focus on SNP sites. A SNP site that contains two different alleles is called biallelic, a SNP site that contains three different alleles is called triallelic and a SNP site that contains four different alleles is called tetraallelic.

A haplotype is the genetic constitution of a sequence of nucleotides 7. The underlying data that forms a haplotype can be the full DNA sequence in the region, or more commonly the SNP sites in that region 7. Polyploid organisms contain two or more homologous haplotypes.

A genotype describes the conflated data of a set of homologous haplotypes. In other words, an explanation for a genotype is a ploidy-specific number of homologous haplotypes. An unphased genotype is a genotype for which no set of explaining haplotypes is defined. There are, however, many possible sets of haplotypes explaining one given unphased genotype. A phased genotype is a genotype for which at least one set of explaining haplotypes is defined. If for a given site all explaining haplotypes have the same value, then the genotype is said to be homozygous at that side. Otherwise the genotype is said to be heterozygous at that side.

Figure 1 illustrates how the different terms are related to each other.

A SNP site of an individual is a string over the nucleotide alphabet Σ = {A, C, G, T} with size determined by the ploidy of the considered species. A sequence of such SNP sites defines a genotype, with the number of SNP sites as the length of the genotype. Let n denote the number of individuals in the sample, m be the number of SNP sites, and p be the ploidy of the considered species. Furthermore, a specific genotype is denoted by g_i, with 1 ≤ i ≤ n, and for a specific site j, with 1 ≤ j ≤ m, in g_i we use g_{i, j}. Finally, let gi,jl MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdYgaSbaaaaa@3252@ with 1 ≤ l ≤ p, denote the l^th state at site j in genotype i. Given a set G MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NbXFeaaa@3755@ of n genotypes, each of length m, the haplotype inference problem is that of finding a set H MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeaaa@368B@ of not necessarily distinct haplotypes. Furthermore, for each genotype g_i ∈ G MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NbXFeaaa@3755@ there is at least one set of p haplotypes {h¹, ..., h^p} ∈ H MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeaaa@368B@ such that g_i is explained by {h¹, ... h^p}. The values of nucleotides are determined by the number of different alleles at the corresponding SNP site: for a biallelic SNP site the values are {0, 1}, for a triallelic SNP site the values are {0, 1, 2}, while for a tetraallelic SNP site the values are {0, 1, 2, 3}. Thus, a specific haplotype h_k is a string over the alphabet {0, 1, 2, 3}, with 1 ≤ k ≤ |H MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeaaa@368B@|.

Polyploid genotypes are represented by sequences of m vectors, where the vectors encode the SNP sites of the given individual. The vectors are of size p and contain alphabetically sorted elements of the alphabet defined by the corresponding SNP site (SNP site always refers to a whole population). For instance, a tetraallelic SNP site of a tetraploid individual (p = 4) which is homozygous is encoded as a vector (0, 0, 0, 0), (1, 1, 1, 1), (2, 2, 2, 2) or (3, 3, 3, 3) depending on the allele found at the given site of the individual. A tetraallelic SNP site of a tetraploid individual at which two alleles occur twice is encoded as a vector (0, 0, 1, 1), (0, 0, 2, 2), (0, 0, 3, 3), (1, 1, 2, 2), (1, 1, 3, 3) or (2, 2, 3, 3) (in general there are eighteen possible encodings, see Table 1).

A tetraallelic SNP site of a tetraploid individual at which two alleles occur once and a third occurs twice is encoded as a vector (0, 1, 2, 2), (0, 1, 3, 3), (0, 2, 3, 3) or (1, 2, 3, 3) (in fact there are twelve possible encodings, see Table 1). Finally, there is only one possible encoding for a tetraallelic SNP site of a tetraploid individual at which all four alleles occur: (0, 1, 2, 3). Triallelic and biallelic SNP sites are encoded accordingly as presented in Table 1. Then, explanation of a genotype is defined as: if p haplotypes explain an unphased genotype g_i, the p haplotypes and the unphased genotype g_i show the same allele composition at each SNP site (Figure 2).

One of the approaches to the haplotype inference problem is called Haplotype Inference by Pure Parsimony 5. A solution to this problem minimises the total number of distinct haplotypes used. The SAT-based formulation of the HIPP models whether there is a set H MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeaaa@368B@ of r distinct haplotypes, with r = |H MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeaaa@368B@| haplotypes, such that each genotype g_i ∈ G MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NbXFeaaa@3755@ is explained by p haplotypes in H MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeaaa@368B@. The SAT-based algorithm considers increasing sizes for H MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeaaa@368B@, from a lower bound lb to an upper bound ub 7. Trivial lower and upper bounds are, respectively, 1 and pn. The algorithm terminates for a size of H MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeaaa@368B@ for which there are r = |H MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeaaa@368B@| haplotypes such that every genotype in G MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NbXFeaaa@3755@ is explained by p haplotypes in H MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeaaa@368B@. The smallest r for which such a set H MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83cHGeaaa@368B@ exists is a most parsimonious set of explaining haplotypes.

All variables of the Boolean satisfiability problem are two-valued. Depending on the truth assignment, a Boolean formula is either true or false. Then, SAT consists of the determination if an assignment to a given Boolean formula in conjunctive normal form (CNF) exists such that the formula evaluates to true, or the proof that such an assignment does not exist. Solving SAT is NP-complete 6. The Boolean satisfiability problem for HIPP, however, can efficiently be solved 7 by SAT solvers such as MiniSat 1112, MiraXT 13 or Sat4J 14. This may be explained by a unknown hidden structure in the genotype data, which makes the problem easier to solve.

The first SAT formulation for HIPP was introduced in 7 and the presented constraints were implemented in the software SHIPs 15. Unfortunately, this approach is restricted to diploid and biallelic species. Here, we extend the formulation of constraints from 7 to polyploid biallelic populations of genotypes. In a tetraploid, biallelic population of genotypes, the possible alleles are modeled by 0 or 1 respectively (e.g. SNP site j of individual i with g_{i, j} = (0, 1, 1, 1), gi,j1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabigdaXaaaaaa@31E1@ = 0, gi,j2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabikdaYaaaaaa@31E3@ = 1, gi,j3 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabiodaZaaaaaa@31E5@ = 1 and gi,j4 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabisda0aaaaaa@31E7@ = 1). Furthermore, the haplotypes can be modeled such that h_{k, j} ∈ {0, 1}, where h_{k, j} denotes the j^th site of haplotype k. A haplotype h_k can then be viewed as a binary word h_k,1 ... h_{k, m} of length m over the alphabet {0, 1}.

For a given value of r, the model considers r haplotypes and aims at finding p haplotypes (which can possibly represent the same haplotype) with each genotype g_i. As a result, for each genotype g_i, the model uses selector variables for selecting which haplotypes are used for explaining g_i. Since the genotype is to be explained by p haplotypes, the model uses p sets of r selector variables, sk,il MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4Cam3aa0baaSqaaiabdUgaRjabcYcaSiabdMgaPbqaaiabdYgaSbaaaaa@326C@. Hence, genotype g_i is explained by haplotypes hk1,...,hkp MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiAaG2aaSbaaSqaaiabdUgaRnaaBaaameaacqaIXaqmaeqaaaWcbeaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiabdIgaOnaaBaaaleaacqWGRbWAdaWgaaadbaGaemiCaahabeaaaSqabaaaaa@38DC@,if sk1,i1=1,...,skp,ip=1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4Cam3aa0baaSqaaiabdUgaRnaaBaaameaacqaIXaqmaeqaaSGaeiilaWIaemyAaKgabaGaeGymaedaaOGaeyypa0JaeGymaeJaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIaem4Cam3aa0baaSqaaiabdUgaRnaaBaaameaacqWGWbaCaeqaaSGaeiilaWIaemyAaKgabaGaemiCaahaaOGaeyypa0JaeGymaedaaa@43CF@.

If the sum of the elements of binary vector g_{i, j} equals 0, then gi,jl MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdYgaSbaaaaa@3252@ = 0, with 1 ≤ l ≤ p. Then the model requires that the following is satisfied:

( ¬ h k , j ∨ ¬ s k , i l ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeiikaGIaeyiRaGTaemiAaG2aaSbaaSqaaiabdUgaRjabcYcaSiabdQgaQbqabaGccqGHOiI2cqGHSca2cqWGZbWCdaqhaaWcbaGaem4AaSMaeiilaWIaemyAaKgabaGaemiBaWgaaOGaeiykaKIaeiilaWcaaa@4051@

where 1 ≤ k ≤ r and 1 ≤ l ≤ p. Hence, if haplotype k is selected for explaining genotype i, by at least one of the p representatives, then the value of haplotype k at site j must be 0. If the sum of the elements of binary vector g_{i, j} equals p, then gi,jl MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdYgaSbaaaaa@3252@ = 1, with 1 ≤ l ≤ p. Then the model requires that the following is satisfied:

( h k , j ∨ ¬ s k , i l ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeiikaGIaemiAaG2aaSbaaSqaaiabdUgaRjabcYcaSiabdQgaQbqabaGccqGHOiI2cqGHSca2cqWGZbWCdaqhaaWcbaGaem4AaSMaeiilaWIaemyAaKgabaGaemiBaWgaaOGaeiykaKIaeiilaWcaaa@3E41@

where 1 ≤ k ≤ r and 1 ≤ l ≤ p. Hence, if haplotype k is selected for explaining genotype i, by at least one of the p representatives, then the value of haplotype k at site j must be 1.

Otherwise, if the sum of the elements of binary vector g_{i, j} does not equal 0 nor p, one requires that the haplotypes explaining genotype g_i show the corresponding number of 1s and 0s at site j. This is achieved by creating p variables gi,j1,...,gi,jp MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabigdaXaaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiabdEgaNnaaDaaaleaacqWGPbqAcqGGSaalcqWGQbGAaeaacqWGWbaCaaaaaa@3CDC@ where gi,jl MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdYgaSbaaaaa@3252@ ∈ {0, 1}, which represent the possible arrangements of 1s and 0s at site j. In the diploid situation, the model requires two clauses in CNF:

( g i , j 1 ∨ g i , j 2 ) ∧ ( ¬ g i , j 1 ∨ ¬ g i , j 2 ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeiikaGIaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabigdaXaaakiabgIIiAlabdEgaNnaaDaaaleaacqWGPbqAcqGGSaalcqWGQbGAaeaacqaIYaGmaaGccqGGPaqkcqGHNis2cqGGOaakcqGHSca2cqWGNbWzdaqhaaWcbaGaemyAaKMaeiilaWIaemOAaOgabaGaeGymaedaaOGaeyikIOTaeyiRaGTaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabikdaYaaakiabcMcaPiabc6caUaaa@51F5@

Formula 3 evaluates to true iff one of the possible allele arrangements of a heterozygous and diploid SNP site is assigned to the Boolean variables. Thus, the formula is equivalent to the enumeration of all possible allele arrangements at the given SNP site, where an arrangement is given as conjunction of literals and all arrangements are connected by disjunctions (disjunctive normal form):

( ¬ g i , j 1 ∧ g i , j 2 ) ∨ ( g i , j 1 ∧ ¬ g i , j 2 ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeiikaGIaeyiRaGTaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabigdaXaaakiabgEIizlabdEgaNnaaDaaaleaacqWGPbqAcqGGSaalcqWGQbGAaeaacqaIYaGmaaGccqGGPaqkcqGHOiI2cqGGOaakcqWGNbWzdaqhaaWcbaGaemyAaKMaeiilaWIaemOAaOgabaGaeGymaedaaOGaey4jIKTaeyiRaGTaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabikdaYaaakiabcMcaPiabc6caUaaa@51F3@

For p > 2 it is also straightforward to formulate the corresponding constraints in a disjunctive normal form (DNF) by enumerating all allele arrangements analogously to Formula 4. Each formula in DNF can be transformed into an equivalent formula in CNF 16. As a result, for the case where g_{i, j} is a heterozygous SNP site, then the model requires that the following is satisfied:

( h k , j ∨ ¬ g i , j l ∨ ¬ s k , i l ) ∧ ( ¬ h k , j ∨ g i , j l ∨ ¬ s k , i l ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeiikaGIaemiAaG2aaSbaaSqaaiabdUgaRjabcYcaSiabdQgaQbqabaGccqGHOiI2cqGHSca2cqWGNbWzdaqhaaWcbaGaemyAaKMaeiilaWIaemOAaOgabaGaemiBaWgaaOGaeyikIOTaeyiRaGTaem4Cam3aa0baaSqaaiabdUgaRjabcYcaSiabdMgaPbqaaiabdYgaSbaakiabcMcaPiabgEIizlabcIcaOiabgYkaylabdIgaOnaaBaaaleaacqWGRbWAcqGGSaalcqWGQbGAaeqaaOGaeyikIOTaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdYgaSbaakiabgIIiAlabgYkaylabdohaZnaaDaaaleaacqWGRbWAcqGGSaalcqWGPbqAaeaacqWGSbaBaaGccqGGPaqkcqGGSaalaaa@65BB@

where 1 ≤ k ≤ r and 1 ≤ l ≤ p. For each i and l, it is necessary that exactly one haplotype is used, and so exactly one selector variable be assigned value 1. For 1 ≤ l ≤ p, this can be captured with cardinality constraints:

( ∑ k = 1 r s k , i l = 1 ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaeWaaeaadaaeWbqaaiabdohaZnaaDaaaleaacqWGRbWAcqGGSaalcqWGPbqAaeaacqWGSbaBaaGccqGH9aqpcqaIXaqmaSqaaiabdUgaRjabg2da9iabigdaXaqaaiabdkhaYbqdcqGHris5aaGccaGLOaGaayzkaaGaeiOla4caaa@3E35@

These sums can be formulated in CNF by the utilization of an additional variables dk,il MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemizaq2aa0baaSqaaiabdUgaRjabcYcaSiabdMgaPbqaaiabdYgaSbaaaaa@324E@ which corresponds to the Boolean state if a haplotype was already selected. The model requires that the following is satisfied:

d 1 , i l ⇔ s 1 , i l ¬ d k , i l ∨ ¬ s k + 1 , i l 2 ≤ k ≤ r − 1 d k + 1 , i l ⇔ ( d k , i l ∨ s k + 1 , i l ) 1 ≤ k ≤ r − 1 d r , i l = 1 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@8818@

where 1 ≤ l ≤ p. Figure 3 illustrates how the variables are used for explaining unphased genotypes.

For the case p > 2, it is straightforward to formulate the constraints for the gi,jl MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdYgaSbaaaaa@3252@ variables from heterozygous SNP sites in DNF by enumerating all allele arrangements. Each formula in DNF can be transformed into an equivalent formula in CNF using Tseitin's transformation 16. However, the enumeration of all arrangements is of exponential complexity. Our objective here is to find an equivalent representation of enumeration of arrangements. This representation is to be in CNF and to allow formulation in polynomial time. Combinatorial problems as described above can also be represented by sums. For instance, for an individual from a tetraploid species with two 0 and two 1 alleles at a biallelic SNP site, all six allele arrangements are determined if the sum of the elements of a binary vector that represents the allele composition is constrained to 2: (0, 0, 1, 1), (0, 1, 0, 1), (1, 0, 0, 1), (1, 0, 1, 0), (1, 1, 0, 0) and (0, 1, 1, 0).

In the following, we combine simple logical circuits, such as half adders and full adders to derive general summation constraints which can be formulated in polynomial time. The usage of one full adder allows the summation of two bits where a full adder consists of two half adders. Formula 8 gives constraints for a half adder, where A and B are two bits which have to be summed, C is the resulting carry over and S_half the resulting sum. Based on a half adder the constraints for a full adder can be derived as given in Formula 9. Variable S_full is the complete sum of the two bits A and B, where the variable C² is the resulting carry over. Variable C¹ represents an additional bit that is added to the sum of A and B. Thus, the total sum of A, B and C¹ is at least 0 and at most 3. The first bit of the result is stored in S_full and the second bit in C².

( A ∧ B ) ⇔ C ¬ ( ( A ∧ B ) ∨ ¬ ( A ∨ B ) ) ⇔ S h a l f MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeWabiqaaaqaaiabcIcaOiabdgeabjabgEIizlabdkeacjabcMcaPiabgsDiBlabdoeadbqaaiabgYkaypaabmaabaGaeiikaGIaemyqaeKaey4jIKTaemOqaiKaeiykaKIaeyikIOTaeyiRaGTaeiikaGIaemyqaeKaeyikIOTaemOqaiKaeiykaKcacaGLOaGaayzkaaGaeyi1HSTaem4uam1aaSbaaSqaaiabdIgaOjabdggaHjabdYgaSjabdAgaMbqabaaaaaaa@5071@

( ( A ∧ B ) ∨ ( S h a l f ∧ C 1 ) ) ⇔ C 2 ( ( ¬ S h a l f ∨ ¬ C 1 ) ∧ ( S h a l f ∨ C 1 ) ) ⇔ S f u l l MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6F12@

The C² carry over of a first full adder may be connected with the C¹ carry over of a second full adder. Analogously, the C² carry over of the second full adder may be connected with a third full adder and so on. If w full adders are connected in this way, the result represents a ripple carry adder that is able to sum up two w bit words. To describe the necessary w carry overs, the model requires that the following is satisfied:

( ( A t ∧ B t ) ∨ ( S h a l f t ∧ C t ) ) ⇔ C t + 1 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaeWaaeaacqGGOaakcqWGbbqqdaahaaWcbeqaaiabdsha0baakiabgEIizlabdkeacnaaCaaaleqabaGaemiDaqhaaOGaeiykaKIaeyikIOTaeiikaGIaem4uam1aa0baaSqaaiabdIgaOjabdggaHjabdYgaSjabdAgaMbqaaiabdsha0baakiabgEIizlabdoeadnaaCaaaleqabaGaemiDaqhaaOGaeiykaKcacaGLOaGaayzkaaGaeyi1HSTaem4qam0aaWbaaSqabeaacqWG0baDcqGHRaWkcqaIXaqmaaGccqGGSaalaaa@4E31@

where t = 1, ..., w - 1. Analogously, to describe the necessary w full adders, the model requires that the following is satisfied:

( ( ¬ S h a l f t ∨ ¬ C t ) ∧ ( S h a l f t ∨ C t ) ) ⇔ S f u l l t , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaeWaaeaacqGGOaakcqGHSca2cqWGtbWudaqhaaWcbaGaemiAaGMaemyyaeMaemiBaWMaemOzaygabaGaemiDaqhaaOGaeyikIOTaeyiRaGTaem4qam0aaWbaaSqabeaacqWG0baDaaGccqGGPaqkcqGHNis2cqGGOaakcqWGtbWudaqhaaWcbaGaemiAaGMaemyyaeMaemiBaWMaemOzaygabaGaemiDaqhaaOGaeyikIOTaem4qam0aaWbaaSqabeaacqWG0baDaaGccqGGPaqkaiaawIcacaGLPaaacqGHuhY2cqWGtbWudaqhaaWcbaGaemOzayMaemyDauNaemiBaWMaemiBaWgabaGaemiDaqhaaOGaeiilaWcaaa@5BAB@

where t = 1, ..., w. The number of necessary bits is at most w = ⌊log₂ p⌋ + 1 if the binary vector which is summed up has length p. Thus, p + 1 different A vectors and p different B, C and S_full vectors are needed. Furthermore, the Sfulll,t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4uam1aa0baaSqaaiabdAgaMjabdwha1jabdYgaSjabdYgaSbqaaiabdYgaSjabcYcaSiabdsha0baaaaa@366D@ variables are replaced by the A^{l+1, t} variables such that only the A^{p+1, t} vector is left for constraining the sum, where 1 ≤ l ≤ p and 1 ≤ t ≤ w. It follows that summing can be represented as shown in Figure 4.

A simplification is achieved if vector C^l,1, where 1 ≤ l ≤ p, (Figure 4) contains the possible allele arrangements 17 and the A vectors store the accumulation of the sum. In this situation, all B variables can be set to zero. The constraints of carry overs reduce to:

(A^{l, t} ∧ C^{l, t}) ⇔ C^{l, t+1},

where 1 ≤ l ≤ p. Additionally, if Sfulll,t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4uam1aa0baaSqaaiabdAgaMjabdwha1jabdYgaSjabdYgaSbqaaiabdYgaSjabcYcaSiabdsha0baaaaa@366D@ variables are replaced by A^{l+1, t} variables, the sums reduce to:

((¬A^{l, t} ∨ ¬C^{l, t}) ∧ (A^{l, t} ∧ C^{l, t})) ⇔ A^{l+1, t},

where 1 ≤ l ≤ p. Reformulating the constraints from Formula 12 into CNF yields the following for the carry overs:

( ¬ A l , t ∨ ¬ C l , t ∨ C l , t + 1 ) ∧ ( ¬ C l , t + 1 ∨ A l , t ) ∧ ( ¬ C l , t + 1 ∨ C l , t ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6CFC@

By reformulating the constraints from Formula 13 into CNF, we obtain the following expression for the sums:

( ¬ A l , t ∨ ¬ C l , t ∨ ¬ A l + 1 , t ) ∧ ( ¬ A l , t ∨ C l , t ∨ A l + 1 , t ) ∧ ( A l , t ∨ ¬ C l , t ∨ A l + 1 , t ) ∧ ( A l , t ∨ C l , t ∨ ¬ A l + 1 , t ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@95AD@

For each individual and SNP site in a biallelic population, variables corresponding to A and C need to be defined. Let variables ai,jl,t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyyae2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdYgaSjabcYcaSiabdsha0baaaaa@3497@, with 1 ≤ l ≤ p + 1 and 1 ≤ t ≤ w, denote the accumulation of the sum.

Additionally, let variables ci,jl,t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4yam2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdYgaSjabcYcaSiabdsha0baaaaa@349B@, with 1 ≤ l ≤ p and 1 ≤ t ≤ w, stand for the carry overs. For a SNP site j from genotype i, summing constraints can easily be obtained by replacing the ci,jl,1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4yam2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdYgaSjabcYcaSiabigdaXaaaaaa@341A@ variables with the gi,jl MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdYgaSbaaaaa@3252@ variables 17, with 1 ≤ l ≤ p. Finally, the ai,jp+1,t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyyae2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdchaWjabgUcaRiabigdaXiabcYcaSiabdsha0baaaaa@3671@ variables, with 1 ≤ t ≤ w, are constrained to the binary representation of the required sum.

In this and the following sections we use notation (x)2y MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeiikaGIaemiEaGNaeiykaKYaa0baaSqaaiabikdaYaqaaiabdMha5baaaaa@319A@ to mean the y^th bit of the binary encoded number x. Furthermore, we use a function b(A, B) to substitute variables:

b ( A , B ) = { A if  B = 1 ¬ A , otherwise . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOyaiMaeiikaGIaemyqaeKaeiilaWIaemOqaiKaeiykaKIaeyypa0ZaaiqaaeaafaqaaeGacaaabaGaemyqaeeabaGaeeyAaKMaeeOzayMaeeiiaaIaemOqaiKaeyypa0JaeGymaedabaGaeyiRaGTaemyqaeKaeiilaWcabaGaee4Ba8MaeeiDaqNaeeiAaGMaeeyzauMaeeOCaiNaee4DaCNaeeyAaKMaee4CamNaeeyzauMaeiOla4caaaGaay5Eaaaaaa@4D1C@

Dependent on the input, Function 16 defines if the substituted variable is negated.

The gi,jl MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdYgaSbaaaaa@3252@ variables are insufficient for describing arrangements of more than two alleles at a SNP site. The representation of, for instance, three different states needs at least two bits. We define o_j as the number of different alleles from all individuals at SNP site j of a population of genotypes. If o_j > 2, the representation of the SNP site is extended to w_j = ⌈log₂ o_j⌉ binary columns. Thus, gi,jl MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdYgaSbaaaaa@3252@ is split to gi,jl,1,...,gi,jl,wj MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdYgaSjabcYcaSiabigdaXaaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiabdEgaNnaaDaaaleaacqWGPbqAcqGGSaalcqWGQbGAaeaacqWGSbaBcqGGSaalcqWG3bWDdaWgaaadbaGaemOAaOgabeaaaaaaaa@42F6@. Each allele is encoded by its corresponding binary number. For instance, the four alleles 0, 1, 2 and 3 at a tetraallelic SNP site are encoded as 00, 01, 10 and 11, respectively. The haplotypes are then extended analogously, such that hk,j∈{0,1}wj MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiAaG2aaSbaaSqaaiabdUgaRjabcYcaSiabdQgaQbqabaGccqGHiiIZcqGG7bWEcqaIWaamcqGGSaalcqaIXaqmcqGG9bqFdaahaaWcbeqaaiabdEha3naaBaaameaacqWGQbGAaeqaaaaaaaa@3B70@ denotes the j^th site of haplotype k.

For generalisation to polyallelic SNP sites, the formulation of binary sums can be reused. Let zi,juj MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOEaO3aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdwha1naaBaaameaacqWGQbGAaeqaaaaaaaa@3414@ be the number of allele u_j at a specific SNP site j in unphased genotype i, where 1 ≤ u_j ≤ o_j. The value of o_j can be greater than p but ∑uj=1ojzi,juj=p MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaabmaeaacqWG6bGEdaqhaaWcbaGaemyAaKMaeiilaWIaemOAaOgabaGaemyDau3aaSbaaWqaaiabdQgaQbqabaaaaOGaeyypa0JaemiCaahaleaacqWG1bqDdaWgaaadbaGaemOAaOgabeaaliabg2da9iabigdaXaqaaiabd+gaVnaaBaaameaacqWGQbGAaeqaaaqdcqGHris5aaaa@407E@. For a set of nucleotide sequences, it holds that o_j ≤ 4.

For each individual and SNP site, o_j vectors v_{i, j} of length p are defined. Variable vi,jl,uj MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemODay3aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdYgaSjabcYcaSiabdwha1naaBaaameaacqWGQbGAaeqaaaaaaaa@364D@ is set to 1 if allele u_j is represented at position l by its corresponding binary number:

( b ( g i , j l , 1 , ( u j − 1 ) 1 2 ) ∧ b ( g i , j l , 2 , ( u j − 1 ) 2 2 ) ∧ ⋮ b ( g i , j l , w j , ( u j − 1 ) 2 w j ) ) ⇔ v i , j l , u j , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@82EB@

where 1 ≤ l ≤ p and 1 ≤ u_j ≤ o_j. It is necessary to formulate the sums ∑l=1pvi,jl,uj MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaabmaeaacqWG2bGDdaqhaaWcbaGaemyAaKMaeiilaWIaemOAaOgabaGaemiBaWMaeiilaWIaemyDau3aaSbaaWqaaiabdQgaQbqabaaaaaWcbaGaemiBaWMaeyypa0JaeGymaedabaGaemiCaahaniabggHiLdaaaa@3D0F@ which must equal zi,juj MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOEaO3aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdwha1naaBaaameaacqWGQbGAaeqaaaaaaaa@3414@, as described in the previous sections such that each allele at SNP site j occurs zi,juj MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOEaO3aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdwha1naaBaaameaacqWGQbGAaeqaaaaaaaa@3414@ times in genotype i.

An example is shown in Figure 5. Generally, at a given SNP site of an individual, there can be at most p different alleles, even if the number of alleles in the population at this site is o_j > p. Thus, only the constraints for p sums have to be given.

In variables vi,jl,uj MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemODay3aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdYgaSjabcYcaSiabdwha1naaBaaameaacqWGQbGAaeqaaaaaaaa@364D@, with 1 ≤ l ≤ p, the arrangements of allele u_j are represented, where allele u_j is encoded as 1 and all other alleles are encoded as 0s. In contrast, the gi,jl,tj MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4zaC2aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdYgaSjabcYcaSiabdsha0naaBaaameaacqWGQbGAaeqaaaaaaaa@362D@ variables, with 1 ≤ l ≤ p and 1 ≤ t_j ≤ w_j, represent the arrangements of all alleles at the given SNP site. It is still necessary that the correct haplotypes hk1,...,hkp MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiAaG2aaSbaaSqaaiabdUgaRnaaBaaameaacqaIXaqmaeqaaaWcbeaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiabdIgaOnaaBaaaleaacqWGRbWAdaWgaaadbaGaemiCaahabeaaaSqabaaaaa@38DC@ are chosen to explain the alleles at g_{i, j}. Then the model requires that the following is satisfied:

( h k , j t j ∨ ¬ g i , j l , t j ∨ ¬ s k , i l ) ∧ ( ¬ h k , j t j ∨ g i , j l , t j ∨ ¬ s k , i l ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7384@

where 1 ≤ t_j ≤ w_j, 1 ≤ k ≤ r and 1 ≤ l ≤ p.

The final (most parsimonious) number of haplotypes is denoted by r_f and the maximal number of alleles, which is found at a SNP site of the considered population, is denoted by o_max. Then, if p log₂ p ≤ r_f log₂ o_max, the number of variables (Table 2) and constraints in the proposed model is, respectively, O MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NdX=eaaa@3765@(n m p² log₂ p) and O MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NdX=eaaa@3765@(r_f n m p log₂ o_max) . The complexity for the variables and constraints decreases to O MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NdX=eaaa@3765@(n m p log₂ p) and O MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NdX=eaaa@3765@(r_f n m p) if the nucleotide alphabet is considered.

It is important to note that the model proposed above is not practical for most existing problem instances, even with the most efficient SAT solvers 7. This problem, however, can be solved by breaking symmetries to prune the search space. As described in 78, symmetries in explaining haplotypes can be broken by sorting the haplotypes lexicographically. A strict lexicographic ordering can be achieved by the formulation of constraints that become true if h₁ is strictly smaller than h₂, h₂ is strictly smaller than h₃, and so on. If the ordering is not strict it is not guaranteed that all explaining haplotypes are pairwise distinct.

For the ordering, h_{k, j} is compared with h_{k+1, j}, where 1 ≤ k ≤ r - 1 and 1 ≤ j ≤ m. Additionally, new variables e_{k, j} have to be introduced, which record the value of h_{k, j} <h_{k+1, j}. Then, the model requires that the following is satisfied:

( ¬ h k , m ∧ h k + 1 , m ) ⇔ e k , m ∧ ( ( ¬ h k , j ∧ h k + 1 , j ) ∨ e k , j + 1 ) ⇔ e k , j ∧ e k , 1 = 1 , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@73A1@

where 1 ≤ j ≤ m + 1. If an e_{k, j} becomes true because the constraint h_{k, j} <h_{k+1, j} is satisfied, it is not necessary to compare h_{k, j'} to h_{k+1, j'}, where j' <j. We must, however, ensure that h_{k, j'} ≤ h_{k+1, j'}, where j' > j. Then, the model requires that the following is satisfied:

(¬h_{k, j} ∨ h_{k+1, j} ∨ e_{k, j}),

where 1 ≤ j ≤ m. If an assignment can be found such that all clauses in Formulas 19 – 20 are true, where 1 ≤ k ≤ r - 1, the haplotypes are in lexicographical order.

Haplotypes that infer a genotype can be lexicographically ordered in a way similar to the set of r explaining haplotypes 78. The l^th haplotype inferring an unphased genotype i is marked by a binary variable sk,il MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4Cam3aa0baaSqaaiabdUgaRjabcYcaSiabdMgaPbqaaiabdYgaSbaaaaa@326C@. The sum ∑k=0rsk,il MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaabmaeaacqWGZbWCdaqhaaWcbaGaem4AaSMaeiilaWIaemyAaKgabaGaemiBaWgaaaqaaiabdUgaRjabg2da9iabicdaWaqaaiabdkhaYbqdcqGHris5aaaa@3923@ is constrained to equal 1 so that exactly one haplotype is selected for explaining the l^th row of g_i. In contrast to the most parsimonious set of explaining haplotypes, the selection variables have to be ordered non-strict lexicographically since homozygous genotypes can not be explained by sets of pairwise distinct haplotypes.

As a result, the p selection vectors of each genotype g_i can be ordered lexicographically by constraining the vectors by si1≤si2≤…≤sip−1≤sip MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4Cam3aa0baaSqaaiabdMgaPbqaaiabigdaXaaakiabgsMiJkabdohaZnaaDaaaleaacqWGPbqAaeaacqaIYaGmaaGccqGHKjYOcqWIMaYscqGHKjYOcqWGZbWCdaqhaaWcbaGaemyAaKgabaGaemiCaaNaeyOeI0IaeGymaedaaOGaeyizImQaem4Cam3aa0baaSqaaiabdMgaPbqaaiabdchaWbaaaaa@4656@78. For this purpose, the constraints of ordering the haplotypes are reused and slightly changed. New variables fk,il MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOzay2aa0baaSqaaiabdUgaRjabcYcaSiabdMgaPbqaaiabdYgaSbaaaaa@3252@ have to be introduced, which record the value of sk,il<sk,il+1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4Cam3aa0baaSqaaiabdUgaRjabcYcaSiabdMgaPbqaaiabdYgaSbaakiabgYda8iabdohaZnaaDaaaleaacqWGRbWAcqGGSaalcqWGPbqAaeaacqWGSbaBcqGHRaWkcqaIXaqmaaaaaa@3BE3@. Then, the model requires that the following is satisfied:

( ¬ s r , i l ∧ s r , i l + 1 ) ⇔ f r , i l ∧ ( ( ¬ s k , i l ∧ s k , i l + 1 ) ∨ f k + 1 , i l ) ⇔ f k , i l , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@754D@

where 1 ≤ k ≤ r + 1. If an fk,il MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOzay2aa0baaSqaaiabdUgaRjabcYcaSiabdMgaPbqaaiabdYgaSbaaaaa@3252@ becomes true because the constraint sk,il<sk,il+1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4Cam3aa0baaSqaaiabdUgaRjabcYcaSiabdMgaPbqaaiabdYgaSbaakiabgYda8iabdohaZnaaDaaaleaacqWGRbWAcqGGSaalcqWGPbqAaeaacqWGSbaBcqGHRaWkcqaIXaqmaaaaaa@3BE3@ is satisfied, it is not necessary to compare sk′,il MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4Cam3aa0baaSqaaiqbdUgaRzaafaGaeiilaWIaemyAaKgabaGaemiBaWgaaaaa@3278@ to sk′,il+1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4Cam3aa0baaSqaaiqbdUgaRzaafaGaeiilaWIaemyAaKgabaGaemiBaWMaey4kaSIaeGymaedaaaaa@344A@, where k' <k. We must, however, ensure that sk′,il≤sk′,il+1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4Cam3aa0baaSqaaiqbdUgaRzaafaGaeiilaWIaemyAaKgabaGaemiBaWgaaOGaeyizImQaem4Cam3aa0baaSqaaiqbdUgaRzaafaGaeiilaWIaemyAaKgabaGaemiBaWMaey4kaSIaeGymaedaaaaa@3CAC@, where k' > k. Then, the model requires that the following is satisfied:

( ¬ s k , i l ∨ s k , i l + 1 ∨ f k , i l ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeiikaGIaeyiRaGTaem4Cam3aa0baaSqaaiabdUgaRjabcYcaSiabdMgaPbqaaiabdYgaSbaakiabgIIiAlabdohaZnaaDaaaleaacqWGRbWAcqGGSaalcqWGPbqAaeaacqWGSbaBcqGHRaWkcqaIXaqmaaGccqGHOiI2cqWGMbGzdaqhaaWcbaGaem4AaSMaeiilaWIaemyAaKgabaGaemiBaWgaaOGaeiykaKIaeiilaWcaaa@49C0@

where 1 ≤ k ≤ r. Because it is almost the formulation of a strict lexicographic order, except that the variable f1,il MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOzay2aa0baaSqaaiabigdaXiabcYcaSiabdMgaPbqaaiabdYgaSbaaaaa@31E3@ does not have to be true, it has to be relaxed to become a non-strict order. This can be done by formulating the constraints for either vector sil MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4Cam3aa0baaSqaaiabdMgaPbqaaiabdYgaSbaaaaa@302D@ is strict smaller than vector sil+1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4Cam3aa0baaSqaaiabdMgaPbqaaiabdYgaSjabgUcaRiabigdaXaaaaaa@31FF@ or both vectors are equal.

Finally, the model requires that the following is satisfied:

( s k , i l ⇔ s k , i l + 1 ) ∨ f 1 , i l , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeiikaGIaem4Cam3aa0baaSqaaiabdUgaRjabcYcaSiabdMgaPbqaaiabdYgaSbaakiabgsDiBlabdohaZnaaDaaaleaacqWGRbWAcqGGSaalcqWGPbqAaeaacqWGSbaBcqGHRaWkcqaIXaqmaaGccqGGPaqkcqGHOiI2cqWGMbGzdaqhaaWcbaGaeGymaeJaeiilaWIaemyAaKgabaGaemiBaWgaaOGaeiilaWcaaa@47ED@

where 1 ≤ k ≤ r. If an assignment can be found for which all clauses in Formulas 21 – 23 are true, where 1 ≤ l ≤ p - 1, the selection variables of genotype g_i are in non-strict lexicographic order.

There can be several most parsimonious sets of haplotypes which differ slightly and yet can explain the unphased genotype data. Call Hk,ji MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemisaG0aa0baaSqaaiabdUgaRjabcYcaSiabdQgaQbqaaiabbMgaPbaaaaa@3210@ the set of previously found binary assignments to h_{k, j}, where 1 ≤ i ≤ N_alt and N_alt is the number of previously found inferences. Constraints for alternative most parsimonious sets of haplotypes can be easily formulated by the exclusion of previously found sets. Then, the model requires that the following is satisfied:

¬ ( b ( h k , 1 , H k , 1 i ) ∧ b ( h k , 2 , H k , 2 i ) ∧ ⋮ b ( h k , m , H k , m i ) ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6245@

where 1 ≤ k ≤ r and 1 ≤ i ≤ N_alt. Application of De Morgan's laws to Formula 24 results in Formula 25:

( ¬ b ( h k , 1 , H k , 1 i ) ∨ ¬ b ( h k , 2 , H k , 2 i ) ∨ ⋮ ¬ b ( h k , m , H k , m i ) ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6669@

A nice feature of constraining alternative haplotype inferences is that Formula 25 is automatically in CNF.

Given p explaining haplotypes, there can also be alternative inferences of genotypes. Constraints of alternative genotype inferences are given similarly to the constraints of alternative haplotype inferences. Call Sk,il,j MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4uam1aa0baaSqaaiabdUgaRjabcYcaSiabdMgaPbqaaiabdYgaSjabcYcaSiabbQgaQbaaaaa@3467@ the set of previously found binary assignments to the selection variables sk,il MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4Cam3aa0baaSqaaiabdUgaRjabcYcaSiabdMgaPbqaaiabdYgaSbaaaaa@326C@ of genotype i, where 1 ≤ j ≤ M_alt and M_alt is the number of previously found inferences. Now consider only one previously found assignment to the selection variables of genotype i with Sk1,i,i1=1,...,Skp,i,ip=1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4uam1aa0baaSqaaiabdUgaRnaaBaaameaacqaIXaqmcqGGSaalcqWGPbqAaeqaaSGaeiilaWIaemyAaKgabaGaeGymaedaaOGaeyypa0JaeGymaeJaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIaem4uam1aa0baaSqaaiabdUgaRnaaBaaameaacqWGWbaCcqGGSaalcqWGPbqAaeqaaSGaeiilaWIaemyAaKgabaGaemiCaahaaOGaeyypa0JaeGymaedaaa@47C5@. Since it is sufficient to constrain selection variables to 0, which were set to 1 in a previously computed inference, the already found explanations of all unphased genotypes are excluded by:

¬ ( s k 1 , 1 , 1 1 ∧ … ∧ s k p , 1 , 1 p ∧ s k 1 , 2 , 2 1 ∧ … ∧ s k p , 2 , 2 p ∧ ⋮ s k 1 , n , n 1 ∧ … ∧ s k p , n , n p ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7CA4@

Application of De Morgan's laws to Formula 26 results in Formula 27:

( ¬ s k 1 , 1 , 1 1 ∨ … ∨ ¬ s k p , 1 , 1 p ∨ ¬ s k 1 , 2 , 2 1 ∨ … ∨ ¬ s k p , 2 , 2 p ∨ ⋮ ¬ s k 1 , n , n 1 ∨ … ∨ ¬ s k p , n , n p ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@8704@

Formula 27 is in CNF and no reformulation is necessary. Such clauses have to be given for each previously found genotype inference j.

For constraining alternative genotype inferences, it is very important that genotype symmetries are broken as shown in Section "Constraints for breaking symmetries in genotypes". If symmetries are not broken, and if an assignment to the sk,il MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4Cam3aa0baaSqaaiabdUgaRjabcYcaSiabdMgaPbqaaiabdYgaSbaaaaa@326C@ variables is excluded for a given unphased genotype g_i, the SAT solver can still report an assignment that represents a permutation of the excluded assignment. For instance, vector sil1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4Cam3aa0baaSqaaiabdMgaPbqaaiabdYgaSnaaBaaameaacqaIXaqmaeqaaaaaaaa@314A@ with length r is exchanged by vector sil2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4Cam3aa0baaSqaaiabdMgaPbqaaiabdYgaSnaaBaaameaacqaIYaGmaeqaaaaaaaa@314C@ with length r.

Note that the number of alternative genotype inferences is equal to or greater than the number of alternative most parsimonious sets of haplotypes, since each alternative set of haplotypes defines at least one inference of genotypes. As a result, we do not calculate complete alternative genotype inferences in this study. Instead, we introduce an optimisation method for genotypes, based on explaining haplotypes and bootstrapping (see Section "Bootstrapping" and Section "Optimisation of genotypes").

All possible minimal inferences are treated equally by the SAT approach. It is unlikely that the first haplotype inference found is the most probable one under the assumed model and given data. It is also unlikely that the first haplotype inference is the inference with fewest differences compared to the real data. The question which haplotype inference should be taken for further analysis remains. There must be one or more inferences which are supported better by the input data. To introduce a quality measurement of the haplotypes and alternative inferences which have been calculated, a bootstrapping procedure is introduced as follows. Bootstrapping is widely used (e.g. in phylogenetic reconstruction 18) for estimating properties of an estimator. Those properties are measured when sampling from an approximate distribution. One standard choice for an approximate distribution is the empirical distribution of the observed data. To use the bootstrap to assess the uncertainty of estimates of the phased genotypes, the data should be a series of independently sampled points. Here, we assume that haplotypes are drawn independently from a most parsimonious set of explaining haplotypes which is the base of the population of genotypes. Thus, the independently drawn haplotypes satisfy the independence assumptions of the bootstrap method.

From a set G MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NbXFeaaa@3755@ of n given unphased genotypes, a new set of n unphased genotypes G′ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGaf8NbXFKbauaaaaa@3761@ is sampled by replacement (Figure 6). Sampled sets G′ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGaf8NbXFKbauaaaaa@3761@ are inferred as described before, and each haplotype occurring in the inferences of the genotypes is added to a list. If a haplotype does not appear in the list, the haplotype and the number of its occurrences in the phased genotypes (its count) are added to the list. Otherwise, the former count of the haplotype in the list is increased by that number. For one G′ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGaf8NbXFKbauaaaaa@3761@, this procedure is defined as one bootstrap replicate. Thus, the count of a haplotype, which is equivalent to a frequency, reflects its support by the input data and is defined as the haplotype's score. The process is repeated, and after a given number of bootstrap replicates, the corresponding counts of each haplotype occurring in the genotypes of an alternative haplotype inference are summed up. This sum is used to score the alternative haplotype inferences. The inference with the greatest score is assumed to be the one with the greatest support from the input data given the parsimony criterion.

Based on computed alternative most parsimonious sets of haplotypes and bootstrapping scores, an optimisation of the phased genotypes can be computed (Figure 7). For each most parsimonious set of haplotypes the algorithm therefore computes all alternative inferences of each unphased genotype, independent of the others. As mentioned above, this method only works if symmetries in genotypes are broken (if symmetries are not broken, the algorithm also reports genotype inferences that use same sets of haplotypes in different orders). This procedure results in a list of alternative phased genotypes for each original unphased genotype. Using the bootstrapping approach, each computed genotype is scored by the sum of the bootstrapping scores of its contained haplotypes. Alternatively, without bootstrapping, the frequencies of the haplotypes in the phased genotypes of all computed alternative inferences can be used similar to bootstrapping counts. Finally, from the alternative inference list of each genotype the highest scored genotype is selected in order to replace the former genotype in the inference. This forms a new inference of the original data with equal or better scoring.

The number of all alternative genotype inferences for a given most parsimonious set of haplotypes is the product of the alternative inferences of each genotype. Moreover, the product of the number of alternative genotype inferences from each alternative most parsimonious set of haplotypes is the number of all valid most parsimonious HIPPs.

In contrast to integer linear programming formulations of HIPP 910, the SAT approach is not able to optimise a target function directly. Thus, each possible number of explaining haplotypes has to be tested incrementally starting with r = 1. Methods for the computation of lower and upper bounds 719 can be applied to avoid the iteration until a most parsimonious solution is found. Furthermore, a lower bound can be used for reducing the size of the model 7. Genotypes which only can be explained by distinct sets of haplotypes are called incompatible. Incompatible genotypes can be used for deriving a lower bound such that the size of the model can be reduced by eliminating s variables and corresponding clauses.

In the existing version of SATlotyper, the computation of lower and upper bounds is not implemented. It was found empirically that, if the approach is able to find a most parsimonious set of haplotypes in reasonable time, it is also able to prove the unsatisfiability of smaller sets of haplotypes in reasonable time. Nevertheless, it is not clear how large the increase in solvable instances would be if a calculation of lower and upper bounds were used in haplotype inference of polyploids. The computation of upper and lower bounds according to 719 may be added to SATlotyper in future versions.

To define a standard for the measurement of an inference of haplotypes and corresponding genotypes, we sum up the differences between genotypes from inference and corresponding genotypes from real data. The number of differences is defined as the distance d between both sets.

Call G_i = hk1,...,hkp MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiAaG2aaSbaaSqaaiabdUgaRnaaBaaameaacqaIXaqmaeqaaaWcbeaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiabdIgaOnaaBaaaleaacqWGRbWAdaWgaaadbaGaemiCaahabeaaaSqabaaaaa@38DC@ the inference of an unphased genotype by the SAT approach and G′i={h′k′1,...,h′k′p} MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafm4raCKbauaadaWgaaWcbaGaemyAaKgabeaakiabg2da9iabcUha7jqbdIgaOzaafaWaaSbaaSqaaiqbdUgaRzaafaWaaSbaaWqaaiabigdaXaqabaaaleqaaOGaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIafmiAaGMbauaadaWgaaWcbaGafm4AaSMbauaadaWgaaadbaGaemiCaahabeaaaSqabaGccqGG9bqFaaa@3FD0@ the real data for instance from simulation. Every haplotype h_k in the inference has a counterpart h′k′ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiAaGMbauaadaWgaaWcbaGafm4AaSMbauaaaeqaaaaa@2ED1@ in the real data set. The distance is defined recursively as shown in Formula 28 and Formula 29, where d_ham(h_k, h'_k') is the Hamming distance between h_k and h'_k'.

D ( G , G ′ ) = { 0 , if  G = { } ∨ G ′ = { } h ∈ G ∧ ∀ h ′ ∈ G ′ : d h a m ( h , h ′ ) + D ( G \ h , G ′ \ h ′ ) otherwise MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@744B@

d = min (D(G, G'))

The complexity of calculating d is exponential but for small p this is still possible.

SATlotyper is implemented in Java and realises the constraints described above. Additionally, there are some obvious improvements included in the program, such as converting the vi,jl,uj MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemODay3aa0baaSqaaiabdMgaPjabcYcaSiabdQgaQbqaaiabdYgaSjabcYcaSiabdwha1naaBaaameaacqWGQbGAaeqaaaaaaaa@364D@ vectors, with 1 ≤ l ≤ p, to the corresponding Boolean inverse if min(q, p - q) = p - q, where q is the number of 1s. Another improvement is the enumeration of constraints for a SNP site such that instead of o_j only o_j - 1 sums have to be given if o_j ≤ p.

The SAT approach that we generalised 7 can not optimise a target function directly (but there are efforts to combine ILP and SAT features 12172021). Therefore, the SAT formulation of an assumed number of explaining haplotypes has to be tested for satisfiability by the SAT solver. If it fails, the number of explaining haplotypes is incremented and then tested again. This is repeated until the SAT solver reports satisfiability. For unphased genotypes, given in CSV format, the program generates corresponding constraints and writes the resulting formula in CNF format to the file system. Next, the binary of the corresponding SAT solver is executed with the newly generated CNF file as input. After successful termination of the solver, SATlotyper reads, analyses and reports the output of the solver in XML format (Additional file 1).

SATlotyper is able to execute different SAT solvers and was tested with MiniSat 1112, MiraXT 13 (a multithreaded SAT solver) and Sat4J 14 but can be easily adapted to other solvers accepting standard CNF file format. Access to single Boolean variables is realised by a hash which contains corresponding matrices. This object allows indexing by means of the corresponding keyword, for instance "haplotype", for a given type of variable.

The presented generalisation of the original SAT approach 7 led to the development of SATlotyper, which can infer polyploid and polyallelic input. The SATlotyper algorithm is able to handle incomplete data sets where SNP sites are partly missing, without bringing in unjustified assumptions. For instance, SNP sites are missing when genotypes are heterozygous for alleles with indels (insertions or deletions) that may result in an interruption of analysable sequence data. Unknown sites are marked "N". With the SAT approach, no assumptions are made for individuals that contain SNP sites with no information available, i.e. the formulation of constraints for the corresponding individual and SNP site is omitted. If the formulation of constraints for a site is omitted, the SAT solver uses a set of haplotypes inferred from other unphased genotypes, provided that these haplotypes are compatible with the known sites of the unphased genotype containing missing information. The choice of haplotypes for explaining such a genotype is independent of the alleles that the explaining haplotypes show at the site with missing information.

In order to test SATlotypers performance, we simulated haplotypes comprising six SNP sites for ten tetraploid, biallelic populations with 100 individuals each. For every population, six simulated haplotypes were used as a pool for further simulation. These six different haplotypes of one population were sampled uniformly to generate a population of tetraploid individuals. The alleles of these haplotypes were also sampled uniformly. The simulation resulted in ten data sets of 100 individuals each.

In order to simulate noise in the simulated data sets, data changes were introduced by conversion of a randomly chosen nucleotide of a randomly chosen individual at a randomly chosen SNP site to the other allele at the same site. All manipulated SNP sites were marked and no longer changed. If, however, a change has to be introduced and the random procedure selects an already manipulated individual and site, the noise-procedure is repeated until the needed change is introduced. Homozygous SNP sites were excluded from change on the assumption that these sites are correctly analysed in real data sets. Moreover, heterozygous SNP sites were not changed to homozygous sites. Thus, x% of noise means exactly x% of changed SNP sites in the data set. The ten simulated data sets were modified by noise, where noise was increased from 0% to 10% in steps of 1% resulting in 110 different data sets. Four types of analysis were performed with the simulated data sets (Table 3). These four methods are referred to as Method 1–4 and are computed as follows.

Method 1: for each data set exactly one haplotype inference was calculated.

Method 2: for each data set up to 250 alternative most parsimonious sets of explaining haplotypes and the corresponding haplotype inferences were calculated. Additionally, bootstrapping was performed based on the calculated haplotypes by generation of 250 bootstrapping replicates. Phased genotypes were then scored by the sum of the scores of their constituent haplotypes, and these values were summed up to score complete haplotype inferences (see Section "Implementation"). The best scored haplotype inference was selected without further optimisation with regard to genotype inference.

Method 3: the analysis described in Method 2 was further refined by an optimisation with regard to genotype inference performed for each alternative most parsimonious set of haplotypes (see Section "Implementation").

Method 4: the first haplotype inference was used to optimise the genotype inference. For this purpose, the haplotypes were scored by their frequency in all genotypes of the first haplotype inference. Next, optimisation of the genotypes was carried out as described.

The phased genotypes resulting from all four types of analysis were compared with the corresponding original simulated data set. We first tested whether the original sets of haplotypes, which were used for generating the simulations, could be identified by Method 1, Method 2 and Method 3. Method 4 was left out since different scorings of one haplotype inference do not affect the inferred most parsimonious set of haplotypes. Without added noise, the six original haplotypes could be identified by all three methods. With the addition of noise it was not possible to identify all six original haplotypes in all data sets using the first haplotype inference (Method 1). For three different data sets, this method found fewer than the six original haplotypes (only five out of six were identified in one data set with 6%, 7% and 10% noise). In contrast, all six original haplotypes were correctly inferred by the analyses with bootstrapping (Method 2) and the analyses with bootstrapping and optimisation (Method 3). For all types of analysis, the phased genotypes resulting from each analysis were compared with the original simulated data sets by computing the minimal Hamming distances between an inferred genotype and the corresponding original genotype without noise from the simulation. The minimal Hamming distance was computed as given in Section "Implementation". Based on the minimal Hamming distance, the correctness for all four types of analysis was calculated as follows:

100 ⋅ 600 − minimal Hamming distance 600 − 600 ⋅ 100 − 1 ⋅ noise % , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6D31@

where 600 is the number of SNP sites in the simulation. The denominator in Formula 30 is motivated by the minimal possible number of incorrectly inferred nucleotides compared to the simulated data without noise (on the right of the sum of the denominator). This value is dependent on the noise used and is subtracted from the number of SNP sites in the data set such that the denominator represents the maximal possible number of correctly inferable nucleotides. The number of correctly inferred nucleotides calculated by the SAT approach is given by the numerator. The mean correctness of the ten different data sets was plotted against the noise (Figure 8).

Without noise, all methods gave predictions close to 100% correctness. With noise added the results of the four analysis methods showed an increasing correctness to the original data in the following order: Method 1 < Method 2 < Method 4 < Method 3. This means that Method 3, which is the method with bootstrapping and genotype optimisation, gave the best results for all values of noise. The comparison between Method 2 and Method 1 demonstrated that the application of bootstrapping in order to select the highest scored haplotype inference (Method 2) gives better results than the method without bootstrapping (Method 1). The distributions of nucleotide distances (minimal Hamming distance) from Method 1 and Method 3 for a given amount of noise were compared by the Kruskal-Wallis test with a significance level of 5%. All p-values except for the 0%-noise case were below 0.05, and consequently the null hypothesis of both distributions being the same was rejected. Although the distributions of nucleotide distances from Method 1 and Method 2 were not significantly different, the mean values of the distances of Method 2 were always smaller than those of Method 1.

The performance of SATlotyper was tested using unphased SNP data from the locus BA213c14t7 of Solanum tuberosum. Locus BA213c14t7 corresponds to the sequenced T7-end of the BAC (bacterial artificial chromosome) clone BA213c14 and is located on potato chromosome V between the markers GP21 and GP179 near the R1 gene for resistance to late blight 22 (see Chromosome V in PoMaMo, The Potato Maps and More Database 2324). This intergenic sequence region is characterised by high sequence variability. The BA213c14t7 sequence also includes SNP sites associated with resistance against the parasitic root cyst nematode Globodera pallida 25.

As input to SATlotyper, two sets of unphased SNP data of BA213c14t7 were generated from 194 heterozygous tetraploid potato individuals from two different breeders: 103 individuals from breeder 1 and 91 individuals from breeder 2. The locus was amplified from genomic DNA of the 194 individuals and the SNP allele dosage (0:4, 1:3, 2:2, 3:1 and 4:0) was estimated for twelve biallelic SNP markers based on the sequence trace files (SNP sites 139, 143, 152, 157, 178, 214, 218, 236, 244, 253, 273, 274; Figure 9; Additional file 2). The resulting unphased SNP data were used as input for a SATlotyper analysis, where only one haplotype inference was calculated (Method 1). The number of SNP sites was varied from two to twelve and the running times were determined. In Figure 10 the log of running time is plotted against the number of SNP sites analysed for the two different data sets (breeder 1, breeder 2). Even for twelve SNP sites, the running time was less than 80 seconds for both data sets. Figure 10 demonstrates that the computational complexity grows exponentially with linear increase of the number of SNP sites.

In order to evaluate SATlotyper further, we compared computed haplotypes with experimentally determined haplotypes at the BA213c14t7 locus using a subset of nineteen heterozygous tetraploid individuals out of the two populations described above. We identified the haplotypes for twelve SNP sites both computationally and experimentally. The sequence of the BA213c14t7 locus and the SNP sites analysed are shown in Figure 9.