Haplotype Inference with Boolean Constraint Solving: An Overview
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paper_title: Efficient haplotype inference with pseudo-Boolean optimization

paper_content:

Haplotype inference from genotype data is a key computational problem in bioinformatics, since retrieving directly haplotype information from DNA samples is not feasible using existing technology. One of the methods for solving this problem uses the pure parsimony criterion, an approach known as Haplotype Inference by Pure Parsimony (HIPP). Initial work in this area was based on a number of different Integer Linear Programming (ILP) models and branch and bound algorithms. Recent work has shown that the utilization of a Boolean Satisfiability (SAT) formulation and state of the art SAT solvers represents the most efficient approach for solving the HIPP problem. ::: ::: Motivated by the promising results obtained using SAT techniques, this paper investigates the utilization of modern Pseudo-Boolean Optimization (PBO) algorithms for solving the HIPP problem. The paper starts by applying PBO to existing ILP models. The results are promising, and motivate the development of a new PBO model (RPoly) for the HIPP problem, which has a compact representation and eliminates key symmetries. Experimental results indicate that RPoly outperforms the SAT-based approach on most problem instances, being, in general, significantly more efficient.
---
paper_title: A New Integer Programming Formulation for the Pure Parsimony Problem in Haplotype Analysis

paper_content:

We present a new integer programming formulation for the haplotype inference by pure parsimony (HIPP) problem. Unlike a previous approach to this problem [2], we create an integer program whose size is polynomial in the size of the input. This IP is substantially smaller for moderate-sized instances of the HIPP problem. We also show several additional constraints, based on the input, that can be added to the IP to aid in finding a solution, and show how to find which of these constraints is active for a given instance in efficient time. We present experimental results that show our IP has comparable success to the formulation of Gusfield [2] on moderate-sized problems, though it is is much slower. However, our formulation can sometimes solve substantially larger problems than are practical with Gusfield’s formulation.
---
paper_title: Efficient haplotype inference with boolean satisfiability

paper_content:

Mutation in DNA is the principal cause for differences among human beings, and Single Nucleotide Polymorphisms (SNPs) are the most common mutations. Hence, a fundamental task is to complete a map of haplotypes (which identify SNPs) in the human population. Associated with this effort, a key computational problem is the inference of haplotype data from genotype data, since in practice genotype data rather than haplotype data is usually obtained. Different haplotype inference approaches have been proposed, including the utilization of statistical methods and the utilization of the pure parsimony criterion. The problem of haplotype inference by pure parsimony (HIPP) is interesting not only because of its application to haplotype inference, but also because it is a challenging NP-hard problem, being APX-hard. Recent work has shown that a SAT-based approach is the most efficient approach for the problem of haplotype inference by pure parsimony (HIPP), being several orders of magnitude faster than existing integer linear programming and branch and bound solutions. This paper provides a detailed description of SHIPs, a SAT-based approach for the HIPP problem, and presents comprehensive experimental results comparing SHIPs with all other exact approaches for the HIPP problem. These results confirm that SHIPs is currently the most effective approach for the HIPP problem.
---
paper_title: Integer programming approaches to haplotype inference by pure parsimony

paper_content:

In 2003, Gusfield introduced the Haplotype Inference by Pure Parsimony (HIPP) problem and presented an integer program (IP) that quickly solved many simulated instances of the problem [1]. Although it solved well on small instances, Gusfield's IP can be of exponential size in the worst case. Several authors [2], [3] have presented polynomial-sized IPs for the problem. In this paper, we further the work on IP approaches to HIPP. We extend the existing polynomial-sized IPs by introducing several classes of valid cuts for the IP. We also present a new polynomial-sized IP formulation that is a hybrid between two existing IP formulations and inherits many of the strengths of both. Many problems that are too complex for the exponential-sized formulations can still be solved in our new formulation in a reasonable amount of time. We provide a detailed empirical comparison of these IP formulations on both simulated and real genotype sequences. Our formulation can also be extended in a variety of ways to allow errors in the input or model the structure of the population under consideration.
---
paper_title: Haplotype inference by pure Parsimony

paper_content:

The next high-priority phase of human genomics will involve the development and use of a full Haplotype Map of the human genome [7]. A critical, perhaps dominating, problem in all such efforts is the inference of large-scale SNP-haplotypes from raw genotype SNP data. This is called the Haplotype Inference (HI) problem. Abstractly, input to the HI problem is a set of n strings over a ternary alphabet. A solution is a set of at most 2n strings over the binary alphabet, so that each input string can be "generated" by some pair of the binary strings in the solution. For greatest biological fidelity, a solution should be consistent with, or evaluated by, properties derived from an appropriate genetic model. ::: ::: A natural model, that has been suggested repeatedly is called here the Pure Parsimony model, where the goal is to find a smallest set of binary strings that can generate the n input strings. The problem of finding such a smallest set is called the Pure Parsimony Problem. Unfortunately, the Pure Parsimony problem is NP-hard, and no paper has previously shown how an optimal Pure-parsimony solution can be computed efficiently for problem instances of the size of current biological interest. In this paper, we show how to formulate the Pure-parsimony problem as an integer linear program; we explain how to improve the practicality of the integer programming formulation; and we present the results of extensive experimentation we have done to show the time and memory practicality of the method, and to compare its accuracy against solutions found by the widely used general haplotyping program PHASE. We also formulate and experiment with variations of the Pure-Parsimony criteria, that allow greater practicality. The results are that the Pure Parsimony problem can be solved efficiently in practice for a wide range of problem instances of current interest in biology. Both the time needed for a solution, and the accuracy of the solution, depend on the level of recombination in the input strings. The speed of the solution improves with increasing recombination, but the accuracy of the solution decreases with increasing recombination.
---
paper_title: Haplotype inference by pure Parsimony

paper_content:

The next high-priority phase of human genomics will involve the development and use of a full Haplotype Map of the human genome [7]. A critical, perhaps dominating, problem in all such efforts is the inference of large-scale SNP-haplotypes from raw genotype SNP data. This is called the Haplotype Inference (HI) problem. Abstractly, input to the HI problem is a set of n strings over a ternary alphabet. A solution is a set of at most 2n strings over the binary alphabet, so that each input string can be "generated" by some pair of the binary strings in the solution. For greatest biological fidelity, a solution should be consistent with, or evaluated by, properties derived from an appropriate genetic model. ::: ::: A natural model, that has been suggested repeatedly is called here the Pure Parsimony model, where the goal is to find a smallest set of binary strings that can generate the n input strings. The problem of finding such a smallest set is called the Pure Parsimony Problem. Unfortunately, the Pure Parsimony problem is NP-hard, and no paper has previously shown how an optimal Pure-parsimony solution can be computed efficiently for problem instances of the size of current biological interest. In this paper, we show how to formulate the Pure-parsimony problem as an integer linear program; we explain how to improve the practicality of the integer programming formulation; and we present the results of extensive experimentation we have done to show the time and memory practicality of the method, and to compare its accuracy against solutions found by the widely used general haplotyping program PHASE. We also formulate and experiment with variations of the Pure-Parsimony criteria, that allow greater practicality. The results are that the Pure Parsimony problem can be solved efficiently in practice for a wide range of problem instances of current interest in biology. Both the time needed for a solution, and the accuracy of the solution, depend on the level of recombination in the input strings. The speed of the solution improves with increasing recombination, but the accuracy of the solution decreases with increasing recombination.
---
paper_title: Integer programming approaches to haplotype inference by pure parsimony

paper_content:

In 2003, Gusfield introduced the Haplotype Inference by Pure Parsimony (HIPP) problem and presented an integer program (IP) that quickly solved many simulated instances of the problem [1]. Although it solved well on small instances, Gusfield's IP can be of exponential size in the worst case. Several authors [2], [3] have presented polynomial-sized IPs for the problem. In this paper, we further the work on IP approaches to HIPP. We extend the existing polynomial-sized IPs by introducing several classes of valid cuts for the IP. We also present a new polynomial-sized IP formulation that is a hybrid between two existing IP formulations and inherits many of the strengths of both. Many problems that are too complex for the exponential-sized formulations can still be solved in our new formulation in a reasonable amount of time. We provide a detailed empirical comparison of these IP formulations on both simulated and real genotype sequences. Our formulation can also be extended in a variety of ways to allow errors in the input or model the structure of the population under consideration.
---
paper_title: Efficient haplotype inference with boolean satisfiability

paper_content:

Mutation in DNA is the principal cause for differences among human beings, and Single Nucleotide Polymorphisms (SNPs) are the most common mutations. Hence, a fundamental task is to complete a map of haplotypes (which identify SNPs) in the human population. Associated with this effort, a key computational problem is the inference of haplotype data from genotype data, since in practice genotype data rather than haplotype data is usually obtained. Different haplotype inference approaches have been proposed, including the utilization of statistical methods and the utilization of the pure parsimony criterion. The problem of haplotype inference by pure parsimony (HIPP) is interesting not only because of its application to haplotype inference, but also because it is a challenging NP-hard problem, being APX-hard. Recent work has shown that a SAT-based approach is the most efficient approach for the problem of haplotype inference by pure parsimony (HIPP), being several orders of magnitude faster than existing integer linear programming and branch and bound solutions. This paper provides a detailed description of SHIPs, a SAT-based approach for the HIPP problem, and presents comprehensive experimental results comparing SHIPs with all other exact approaches for the HIPP problem. These results confirm that SHIPs is currently the most effective approach for the HIPP problem.
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paper_title: Efficient and Tight Upper Bounds for Haplotype Inference by Pure Parsimony using Delayed Haplotype Selection

paper_content:

Haplotype inference from genotype data is a key step towards a better understanding of the role played by genetic variations on inherited diseases. One of the most promising approaches uses the pure parsimony criterion. This approach is called Haplotype Inference by Pure Parsimony (HIPP) and is NP-hard as it aims at minimising the number of haplotypes required to explain a given set of genotypes. The HIPP problem is often solved using constraint satisfaction techniques, for which the upper bound on the number of required haplotypes is a key issue. Another very well-known approach is Clark's method, which resolves genotypes by greedily selecting an explaining pair of haplotypes. In this work, we combine the basic idea of Clark's method with a more sophisticated method for the selection of explaining haplotypes, in order to explicitly introduce a bias towards parsimonious explanations. This new algorithm can be used either to obtain an approximated solution to the HIPP problem or to obtain an upper bound on the size of the pure parsimony solution. This upper bound can then used to efficiently encode the problem as a constraint satisfaction problem. The experimental evaluation, conducted using a large set of real and artificially generated examples, shows that the new method is much more effective than Clark's method at obtaining parsimonious solutions, while keeping the advantages of simplicity and speed of Clark's method.
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paper_title: Inference of haplotypes from PCR-amplified samples of diploid populations.

paper_content:

Direct sequencing of genomic DNA from diploid individuals leads to ambiguities on sequencing gels whenever there is more than one mismatching site in the sequences of the two orthologous copies of a gene. While these ambiguities cannot be resolved from a single sample without resorting to other experimental methods (such as cloning in the traditional way), population samples may be useful for inferring haplotypes. For each individual in the sample that is homozygous for the amplified sequence, there are no ambiguities in the identification of the allele’s sequence. The sequences of other alleles can be inferred by taking the remaining sequence after “subtracting off’ the sequencing ladder of each known site. Details of the algorithm for extracting allelic sequences from such data are presented here, along with some population-genetic considerations that influence the likelihood for success of the method. The algorithm also applies to the problem of inferring haplotype frequencies of closely linked restriction-site polymorphisms.
---
paper_title: A New Integer Programming Formulation for the Pure Parsimony Problem in Haplotype Analysis

paper_content:

We present a new integer programming formulation for the haplotype inference by pure parsimony (HIPP) problem. Unlike a previous approach to this problem [2], we create an integer program whose size is polynomial in the size of the input. This IP is substantially smaller for moderate-sized instances of the HIPP problem. We also show several additional constraints, based on the input, that can be added to the IP to aid in finding a solution, and show how to find which of these constraints is active for a given instance in efficient time. We present experimental results that show our IP has comparable success to the formulation of Gusfield [2] on moderate-sized problems, though it is is much slower. However, our formulation can sometimes solve substantially larger problems than are practical with Gusfield’s formulation.
---
paper_title: A survey of computational methods for determining haplotypes

paper_content:

It is widely anticipated that the study of variation in the human genome will provide a means of predicting risk of a variety of complex diseases. Single nucleotide polymorphisms (SNPs) are the most common form of genomic variation. Haplotypes have been suggested as one means for reducing the complexity of studying SNPs. In this paper we review some of the computational approaches that have been taking for determining haplotypes and suggest new approaches.
---
paper_title: Integer programming approaches to haplotype inference by pure parsimony

paper_content:

In 2003, Gusfield introduced the Haplotype Inference by Pure Parsimony (HIPP) problem and presented an integer program (IP) that quickly solved many simulated instances of the problem [1]. Although it solved well on small instances, Gusfield's IP can be of exponential size in the worst case. Several authors [2], [3] have presented polynomial-sized IPs for the problem. In this paper, we further the work on IP approaches to HIPP. We extend the existing polynomial-sized IPs by introducing several classes of valid cuts for the IP. We also present a new polynomial-sized IP formulation that is a hybrid between two existing IP formulations and inherits many of the strengths of both. Many problems that are too complex for the exponential-sized formulations can still be solved in our new formulation in a reasonable amount of time. We provide a detailed empirical comparison of these IP formulations on both simulated and real genotype sequences. Our formulation can also be extended in a variety of ways to allow errors in the input or model the structure of the population under consideration.
---
paper_title: Haplotype inference by pure Parsimony

paper_content:

The next high-priority phase of human genomics will involve the development and use of a full Haplotype Map of the human genome [7]. A critical, perhaps dominating, problem in all such efforts is the inference of large-scale SNP-haplotypes from raw genotype SNP data. This is called the Haplotype Inference (HI) problem. Abstractly, input to the HI problem is a set of n strings over a ternary alphabet. A solution is a set of at most 2n strings over the binary alphabet, so that each input string can be "generated" by some pair of the binary strings in the solution. For greatest biological fidelity, a solution should be consistent with, or evaluated by, properties derived from an appropriate genetic model. ::: ::: A natural model, that has been suggested repeatedly is called here the Pure Parsimony model, where the goal is to find a smallest set of binary strings that can generate the n input strings. The problem of finding such a smallest set is called the Pure Parsimony Problem. Unfortunately, the Pure Parsimony problem is NP-hard, and no paper has previously shown how an optimal Pure-parsimony solution can be computed efficiently for problem instances of the size of current biological interest. In this paper, we show how to formulate the Pure-parsimony problem as an integer linear program; we explain how to improve the practicality of the integer programming formulation; and we present the results of extensive experimentation we have done to show the time and memory practicality of the method, and to compare its accuracy against solutions found by the widely used general haplotyping program PHASE. We also formulate and experiment with variations of the Pure-Parsimony criteria, that allow greater practicality. The results are that the Pure Parsimony problem can be solved efficiently in practice for a wide range of problem instances of current interest in biology. Both the time needed for a solution, and the accuracy of the solution, depend on the level of recombination in the input strings. The speed of the solution improves with increasing recombination, but the accuracy of the solution decreases with increasing recombination.
---
paper_title: Haplotype inference by pure Parsimony

paper_content:

The next high-priority phase of human genomics will involve the development and use of a full Haplotype Map of the human genome [7]. A critical, perhaps dominating, problem in all such efforts is the inference of large-scale SNP-haplotypes from raw genotype SNP data. This is called the Haplotype Inference (HI) problem. Abstractly, input to the HI problem is a set of n strings over a ternary alphabet. A solution is a set of at most 2n strings over the binary alphabet, so that each input string can be "generated" by some pair of the binary strings in the solution. For greatest biological fidelity, a solution should be consistent with, or evaluated by, properties derived from an appropriate genetic model. ::: ::: A natural model, that has been suggested repeatedly is called here the Pure Parsimony model, where the goal is to find a smallest set of binary strings that can generate the n input strings. The problem of finding such a smallest set is called the Pure Parsimony Problem. Unfortunately, the Pure Parsimony problem is NP-hard, and no paper has previously shown how an optimal Pure-parsimony solution can be computed efficiently for problem instances of the size of current biological interest. In this paper, we show how to formulate the Pure-parsimony problem as an integer linear program; we explain how to improve the practicality of the integer programming formulation; and we present the results of extensive experimentation we have done to show the time and memory practicality of the method, and to compare its accuracy against solutions found by the widely used general haplotyping program PHASE. We also formulate and experiment with variations of the Pure-Parsimony criteria, that allow greater practicality. The results are that the Pure Parsimony problem can be solved efficiently in practice for a wide range of problem instances of current interest in biology. Both the time needed for a solution, and the accuracy of the solution, depend on the level of recombination in the input strings. The speed of the solution improves with increasing recombination, but the accuracy of the solution decreases with increasing recombination.
---
paper_title: A New Integer Programming Formulation for the Pure Parsimony Problem in Haplotype Analysis

paper_content:

We present a new integer programming formulation for the haplotype inference by pure parsimony (HIPP) problem. Unlike a previous approach to this problem [2], we create an integer program whose size is polynomial in the size of the input. This IP is substantially smaller for moderate-sized instances of the HIPP problem. We also show several additional constraints, based on the input, that can be added to the IP to aid in finding a solution, and show how to find which of these constraints is active for a given instance in efficient time. We present experimental results that show our IP has comparable success to the formulation of Gusfield [2] on moderate-sized problems, though it is is much slower. However, our formulation can sometimes solve substantially larger problems than are practical with Gusfield’s formulation.
---
paper_title: A survey of computational methods for determining haplotypes

paper_content:

It is widely anticipated that the study of variation in the human genome will provide a means of predicting risk of a variety of complex diseases. Single nucleotide polymorphisms (SNPs) are the most common form of genomic variation. Haplotypes have been suggested as one means for reducing the complexity of studying SNPs. In this paper we review some of the computational approaches that have been taking for determining haplotypes and suggest new approaches.
---
paper_title: Integer programming approaches to haplotype inference by pure parsimony

paper_content:

In 2003, Gusfield introduced the Haplotype Inference by Pure Parsimony (HIPP) problem and presented an integer program (IP) that quickly solved many simulated instances of the problem [1]. Although it solved well on small instances, Gusfield's IP can be of exponential size in the worst case. Several authors [2], [3] have presented polynomial-sized IPs for the problem. In this paper, we further the work on IP approaches to HIPP. We extend the existing polynomial-sized IPs by introducing several classes of valid cuts for the IP. We also present a new polynomial-sized IP formulation that is a hybrid between two existing IP formulations and inherits many of the strengths of both. Many problems that are too complex for the exponential-sized formulations can still be solved in our new formulation in a reasonable amount of time. We provide a detailed empirical comparison of these IP formulations on both simulated and real genotype sequences. Our formulation can also be extended in a variety of ways to allow errors in the input or model the structure of the population under consideration.
---
paper_title: Boosting Haplotype Inference with Local Search

paper_content:

A very challenging problem in the genetics domain is to infer haplotypes from genotypes. This process is expected to identify genes affecting health, disease and response to drugs. One of the approaches to haplotype inference aims to minimise the number of different haplotypes used, and is known as haplotype inference by pure parsimony (HIPP). The HIPP problem is computationally difficult, being NP-hard. Recently, a SAT-based method (SHIPs) has been proposed to solve the HIPP problem. This method iteratively considers an increasing number of haplotypes, starting from an initial lower bound. Hence, one important aspect of SHIPs is the lower bounding procedure, which reduces the number of iterations of the basic algorithm, and also indirectly simplifies the resulting SAT model. This paper describes the use of local search to improve existing lower bounding procedures. The new lower bounding procedure is guaranteed to be as tight as the existing procedures. In practice the new procedure is in most cases considerably tighter, allowing significant improvement of performance on challenging problem instances.
---
paper_title: Efficient haplotype inference with boolean satisfiability

paper_content:

Mutation in DNA is the principal cause for differences among human beings, and Single Nucleotide Polymorphisms (SNPs) are the most common mutations. Hence, a fundamental task is to complete a map of haplotypes (which identify SNPs) in the human population. Associated with this effort, a key computational problem is the inference of haplotype data from genotype data, since in practice genotype data rather than haplotype data is usually obtained. Different haplotype inference approaches have been proposed, including the utilization of statistical methods and the utilization of the pure parsimony criterion. The problem of haplotype inference by pure parsimony (HIPP) is interesting not only because of its application to haplotype inference, but also because it is a challenging NP-hard problem, being APX-hard. Recent work has shown that a SAT-based approach is the most efficient approach for the problem of haplotype inference by pure parsimony (HIPP), being several orders of magnitude faster than existing integer linear programming and branch and bound solutions. This paper provides a detailed description of SHIPs, a SAT-based approach for the HIPP problem, and presents comprehensive experimental results comparing SHIPs with all other exact approaches for the HIPP problem. These results confirm that SHIPs is currently the most effective approach for the HIPP problem.
---
paper_title: Haplotype inference from unphased SNP data in heterozygous polyploids based on SAT

paper_content:

BackgroundHaplotype inference based on unphased SNP markers is an important task in population genetics. Although there are different approaches to the inference of haplotypes in diploid species, the existing software is not suitable for inferring haplotypes from unphased SNP data in polyploid species, such as the cultivated potato (Solanum tuberosum). Potato species are tetraploid and highly heterozygous.ResultsHere we present the software SATlotyper which is able to handle polyploid and polyallelic data. SATlo-typer uses the Boolean satisfiability problem to formulate Haplotype Inference by Pure Parsimony. The software excludes existing haplotype inferences, thus allowing for calculation of alternative inferences. As it is not known which of the multiple haplotype inferences are best supported by the given unphased data set, we use a bootstrapping procedure that allows for scoring of alternative inferences. Finally, by means of the bootstrapping scores, it is possible to optimise the phased genotypes belonging to a given haplotype inference. The program is evaluated with simulated and experimental SNP data generated for heterozygous tetraploid populations of potato. We show that, instead of taking the first haplotype inference reported by the program, we can significantly improve the quality of the final result by applying additional methods that include scoring of the alternative haplotype inferences and genotype optimisation. For a sub-population of nineteen individuals, the predicted results computed by SATlotyper were directly compared with results obtained by experimental haplotype inference via sequencing of cloned amplicons. Prediction and experiment gave similar results regarding the inferred haplotypes and phased genotypes.ConclusionOur results suggest that Haplotype Inference by Pure Parsimony can be solved efficiently by the SAT approach, even for data sets of unphased SNP from heterozygous polyploids. SATlotyper is freeware and is distributed as a Java JAR file. The software can be downloaded from the webpage of the GABI Primary Database at http://www.gabipd.org/projects/satlotyper/. The application of SATlotyper will provide haplotype information, which can be used in haplotype association mapping studies of polyploid plants.
---
paper_title: Efficient Haplotype Inference with Combined CP and OR Techniques

paper_content:

Haplotype inference has relevant biological applications, and represents a challenging computational problem. Among others, pure parsimony provides a viable modeling approach for haplotype inference and provides a simple optimization criterion. Alternative approaches have been proposed for haplotype inference by pure parsimony (HIPP), including branch and bound, integer programming and, more recently, propositional satisfiability and pseudo-Boolean optimization (PBO). Among these, the currently best performing HIPP approach is based on PBO. This paper proposes a number of effective improvements to PBO-based HIPP, including the use of lower bounding and pruning techniques effective with other approaches. The new PBO-based HIPP approach reduces by 50% the number of instances that remain unsolvable by HIPP based approaches.
---
paper_title: Towards robust CNF encodings of cardinality constraints

paper_content:

Motivated by the performance improvements made to SAT solvers in recent years, a number of different encodings of constraints into SAT have been proposed. Concrete examples are the different SAT encodings for ≤ 1 (x1, . . . , xn) constraints. The most widely used encoding is known as the pairwise encoding, which is quadratic in the number of variables in the constraint. Alternative encodings are in general linear, and require using additional auxiliary variables. In most settings, the pairwise encoding performs acceptably well, but can require unacceptably large Boolean formulas. In contrast, linear encodings yield much smaller Boolean formulas, but in practice SAT solvers often perform unpredictably. This lack of predictability is mostly due to the large number of auxiliary variables that need to be added to the resulting Boolean formula. This paper studies one specific encoding for ≤ 1 (x1, . . . , xn) constraints, and shows how a state-of-the-art SAT solver can be adapted to overcome the problem of adding additional auxiliary variables. Moreover, the paper shows that a SAT solver may essentially ignore the existence of auxiliary variables. Experimental results indicate that the modified SAT solver becomes significantly more robust on SAT encodings involving ≤ 1 (x1, . . . , xn) constraints.
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paper_title: Efficient Haplotype Inference with Answer Set Programming

paper_content:

Identifying maternal and paternal inheritance is essential to be able to find the set of genes responsible for a particular disease. However, due to technological limitations, we have access to genotype data (genetic makeup of an individual), and determining haplotypes (genetic makeup of the parents) experimentally is a costly and time consuming procedure. With these biological motivations, we study a computational problem, called Haplotype Inference by Pure Parsimony (HIPP), that asks for the minimal number of haplotypes that form a given set of genotypes. HIPP has been studied using integer linear programming, branch and bound algorithms, SAT-based algorithms, or pseudo-boolean optimization methods. We introduce a new approach to solving HIPP, using Answer Set Programming (ASP). According to our experiments with a large number of problem instances (some automatically generated and some real), our ASP-based approach solves the most number of problems compared with other approaches. Due to the expressivity of the knowledge representation language of ASP, our approach allows us to solve variations of HIPP, e.g., with additional domain specific information, such as patterns/parts of haplotypes observed for some gene family, or with some missing genotype information. In this sense, the ASP-based approach is more general than the existing approaches to haplotype inference.
---
paper_title: Efficient haplotype inference with pseudo-Boolean optimization

paper_content:

Haplotype inference from genotype data is a key computational problem in bioinformatics, since retrieving directly haplotype information from DNA samples is not feasible using existing technology. One of the methods for solving this problem uses the pure parsimony criterion, an approach known as Haplotype Inference by Pure Parsimony (HIPP). Initial work in this area was based on a number of different Integer Linear Programming (ILP) models and branch and bound algorithms. Recent work has shown that the utilization of a Boolean Satisfiability (SAT) formulation and state of the art SAT solvers represents the most efficient approach for solving the HIPP problem. ::: ::: Motivated by the promising results obtained using SAT techniques, this paper investigates the utilization of modern Pseudo-Boolean Optimization (PBO) algorithms for solving the HIPP problem. The paper starts by applying PBO to existing ILP models. The results are promising, and motivate the development of a new PBO model (RPoly) for the HIPP problem, which has a compact representation and eliminates key symmetries. Experimental results indicate that RPoly outperforms the SAT-based approach on most problem instances, being, in general, significantly more efficient.
---
paper_title: Efficient haplotype inference with boolean satisfiability

paper_content:

Mutation in DNA is the principal cause for differences among human beings, and Single Nucleotide Polymorphisms (SNPs) are the most common mutations. Hence, a fundamental task is to complete a map of haplotypes (which identify SNPs) in the human population. Associated with this effort, a key computational problem is the inference of haplotype data from genotype data, since in practice genotype data rather than haplotype data is usually obtained. Different haplotype inference approaches have been proposed, including the utilization of statistical methods and the utilization of the pure parsimony criterion. The problem of haplotype inference by pure parsimony (HIPP) is interesting not only because of its application to haplotype inference, but also because it is a challenging NP-hard problem, being APX-hard. Recent work has shown that a SAT-based approach is the most efficient approach for the problem of haplotype inference by pure parsimony (HIPP), being several orders of magnitude faster than existing integer linear programming and branch and bound solutions. This paper provides a detailed description of SHIPs, a SAT-based approach for the HIPP problem, and presents comprehensive experimental results comparing SHIPs with all other exact approaches for the HIPP problem. These results confirm that SHIPs is currently the most effective approach for the HIPP problem.
---
paper_title: Integer programming approaches to haplotype inference by pure parsimony

paper_content:

In 2003, Gusfield introduced the Haplotype Inference by Pure Parsimony (HIPP) problem and presented an integer program (IP) that quickly solved many simulated instances of the problem [1]. Although it solved well on small instances, Gusfield's IP can be of exponential size in the worst case. Several authors [2], [3] have presented polynomial-sized IPs for the problem. In this paper, we further the work on IP approaches to HIPP. We extend the existing polynomial-sized IPs by introducing several classes of valid cuts for the IP. We also present a new polynomial-sized IP formulation that is a hybrid between two existing IP formulations and inherits many of the strengths of both. Many problems that are too complex for the exponential-sized formulations can still be solved in our new formulation in a reasonable amount of time. We provide a detailed empirical comparison of these IP formulations on both simulated and real genotype sequences. Our formulation can also be extended in a variety of ways to allow errors in the input or model the structure of the population under consideration.
---
paper_title: Efficient Haplotype Inference with Combined CP and OR Techniques

paper_content:

Haplotype inference has relevant biological applications, and represents a challenging computational problem. Among others, pure parsimony provides a viable modeling approach for haplotype inference and provides a simple optimization criterion. Alternative approaches have been proposed for haplotype inference by pure parsimony (HIPP), including branch and bound, integer programming and, more recently, propositional satisfiability and pseudo-Boolean optimization (PBO). Among these, the currently best performing HIPP approach is based on PBO. This paper proposes a number of effective improvements to PBO-based HIPP, including the use of lower bounding and pruning techniques effective with other approaches. The new PBO-based HIPP approach reduces by 50% the number of instances that remain unsolvable by HIPP based approaches.
---
paper_title: Efficient haplotype inference with pseudo-Boolean optimization

paper_content:

Haplotype inference from genotype data is a key computational problem in bioinformatics, since retrieving directly haplotype information from DNA samples is not feasible using existing technology. One of the methods for solving this problem uses the pure parsimony criterion, an approach known as Haplotype Inference by Pure Parsimony (HIPP). Initial work in this area was based on a number of different Integer Linear Programming (ILP) models and branch and bound algorithms. Recent work has shown that the utilization of a Boolean Satisfiability (SAT) formulation and state of the art SAT solvers represents the most efficient approach for solving the HIPP problem. ::: ::: Motivated by the promising results obtained using SAT techniques, this paper investigates the utilization of modern Pseudo-Boolean Optimization (PBO) algorithms for solving the HIPP problem. The paper starts by applying PBO to existing ILP models. The results are promising, and motivate the development of a new PBO model (RPoly) for the HIPP problem, which has a compact representation and eliminates key symmetries. Experimental results indicate that RPoly outperforms the SAT-based approach on most problem instances, being, in general, significantly more efficient.
---
paper_title: A New Integer Programming Formulation for the Pure Parsimony Problem in Haplotype Analysis

paper_content:

We present a new integer programming formulation for the haplotype inference by pure parsimony (HIPP) problem. Unlike a previous approach to this problem [2], we create an integer program whose size is polynomial in the size of the input. This IP is substantially smaller for moderate-sized instances of the HIPP problem. We also show several additional constraints, based on the input, that can be added to the IP to aid in finding a solution, and show how to find which of these constraints is active for a given instance in efficient time. We present experimental results that show our IP has comparable success to the formulation of Gusfield [2] on moderate-sized problems, though it is is much slower. However, our formulation can sometimes solve substantially larger problems than are practical with Gusfield’s formulation.
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paper_title: Efficient Haplotype Inference with Combined CP and OR Techniques

paper_content:

Haplotype inference has relevant biological applications, and represents a challenging computational problem. Among others, pure parsimony provides a viable modeling approach for haplotype inference and provides a simple optimization criterion. Alternative approaches have been proposed for haplotype inference by pure parsimony (HIPP), including branch and bound, integer programming and, more recently, propositional satisfiability and pseudo-Boolean optimization (PBO). Among these, the currently best performing HIPP approach is based on PBO. This paper proposes a number of effective improvements to PBO-based HIPP, including the use of lower bounding and pruning techniques effective with other approaches. The new PBO-based HIPP approach reduces by 50% the number of instances that remain unsolvable by HIPP based approaches.
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paper_title: Efficient and Tight Upper Bounds for Haplotype Inference by Pure Parsimony using Delayed Haplotype Selection

paper_content:

Haplotype inference from genotype data is a key step towards a better understanding of the role played by genetic variations on inherited diseases. One of the most promising approaches uses the pure parsimony criterion. This approach is called Haplotype Inference by Pure Parsimony (HIPP) and is NP-hard as it aims at minimising the number of haplotypes required to explain a given set of genotypes. The HIPP problem is often solved using constraint satisfaction techniques, for which the upper bound on the number of required haplotypes is a key issue. Another very well-known approach is Clark's method, which resolves genotypes by greedily selecting an explaining pair of haplotypes. In this work, we combine the basic idea of Clark's method with a more sophisticated method for the selection of explaining haplotypes, in order to explicitly introduce a bias towards parsimonious explanations. This new algorithm can be used either to obtain an approximated solution to the HIPP problem or to obtain an upper bound on the size of the pure parsimony solution. This upper bound can then used to efficiently encode the problem as a constraint satisfaction problem. The experimental evaluation, conducted using a large set of real and artificially generated examples, shows that the new method is much more effective than Clark's method at obtaining parsimonious solutions, while keeping the advantages of simplicity and speed of Clark's method.
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paper_title: A New Integer Programming Formulation for the Pure Parsimony Problem in Haplotype Analysis

paper_content:

We present a new integer programming formulation for the haplotype inference by pure parsimony (HIPP) problem. Unlike a previous approach to this problem [2], we create an integer program whose size is polynomial in the size of the input. This IP is substantially smaller for moderate-sized instances of the HIPP problem. We also show several additional constraints, based on the input, that can be added to the IP to aid in finding a solution, and show how to find which of these constraints is active for a given instance in efficient time. We present experimental results that show our IP has comparable success to the formulation of Gusfield [2] on moderate-sized problems, though it is is much slower. However, our formulation can sometimes solve substantially larger problems than are practical with Gusfield’s formulation.
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paper_title: Integer programming approaches to haplotype inference by pure parsimony

paper_content:

In 2003, Gusfield introduced the Haplotype Inference by Pure Parsimony (HIPP) problem and presented an integer program (IP) that quickly solved many simulated instances of the problem [1]. Although it solved well on small instances, Gusfield's IP can be of exponential size in the worst case. Several authors [2], [3] have presented polynomial-sized IPs for the problem. In this paper, we further the work on IP approaches to HIPP. We extend the existing polynomial-sized IPs by introducing several classes of valid cuts for the IP. We also present a new polynomial-sized IP formulation that is a hybrid between two existing IP formulations and inherits many of the strengths of both. Many problems that are too complex for the exponential-sized formulations can still be solved in our new formulation in a reasonable amount of time. We provide a detailed empirical comparison of these IP formulations on both simulated and real genotype sequences. Our formulation can also be extended in a variety of ways to allow errors in the input or model the structure of the population under consideration.
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paper_title: Haplotype inference by pure Parsimony

paper_content:

The next high-priority phase of human genomics will involve the development and use of a full Haplotype Map of the human genome [7]. A critical, perhaps dominating, problem in all such efforts is the inference of large-scale SNP-haplotypes from raw genotype SNP data. This is called the Haplotype Inference (HI) problem. Abstractly, input to the HI problem is a set of n strings over a ternary alphabet. A solution is a set of at most 2n strings over the binary alphabet, so that each input string can be "generated" by some pair of the binary strings in the solution. For greatest biological fidelity, a solution should be consistent with, or evaluated by, properties derived from an appropriate genetic model. ::: ::: A natural model, that has been suggested repeatedly is called here the Pure Parsimony model, where the goal is to find a smallest set of binary strings that can generate the n input strings. The problem of finding such a smallest set is called the Pure Parsimony Problem. Unfortunately, the Pure Parsimony problem is NP-hard, and no paper has previously shown how an optimal Pure-parsimony solution can be computed efficiently for problem instances of the size of current biological interest. In this paper, we show how to formulate the Pure-parsimony problem as an integer linear program; we explain how to improve the practicality of the integer programming formulation; and we present the results of extensive experimentation we have done to show the time and memory practicality of the method, and to compare its accuracy against solutions found by the widely used general haplotyping program PHASE. We also formulate and experiment with variations of the Pure-Parsimony criteria, that allow greater practicality. The results are that the Pure Parsimony problem can be solved efficiently in practice for a wide range of problem instances of current interest in biology. Both the time needed for a solution, and the accuracy of the solution, depend on the level of recombination in the input strings. The speed of the solution improves with increasing recombination, but the accuracy of the solution decreases with increasing recombination.
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paper_title: Efficient Haplotype Inference with Combined CP and OR Techniques

paper_content:

Haplotype inference has relevant biological applications, and represents a challenging computational problem. Among others, pure parsimony provides a viable modeling approach for haplotype inference and provides a simple optimization criterion. Alternative approaches have been proposed for haplotype inference by pure parsimony (HIPP), including branch and bound, integer programming and, more recently, propositional satisfiability and pseudo-Boolean optimization (PBO). Among these, the currently best performing HIPP approach is based on PBO. This paper proposes a number of effective improvements to PBO-based HIPP, including the use of lower bounding and pruning techniques effective with other approaches. The new PBO-based HIPP approach reduces by 50% the number of instances that remain unsolvable by HIPP based approaches.
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paper_title: Inference of haplotypes from PCR-amplified samples of diploid populations.

paper_content:

Direct sequencing of genomic DNA from diploid individuals leads to ambiguities on sequencing gels whenever there is more than one mismatching site in the sequences of the two orthologous copies of a gene. While these ambiguities cannot be resolved from a single sample without resorting to other experimental methods (such as cloning in the traditional way), population samples may be useful for inferring haplotypes. For each individual in the sample that is homozygous for the amplified sequence, there are no ambiguities in the identification of the allele’s sequence. The sequences of other alleles can be inferred by taking the remaining sequence after “subtracting off’ the sequencing ladder of each known site. Details of the algorithm for extracting allelic sequences from such data are presented here, along with some population-genetic considerations that influence the likelihood for success of the method. The algorithm also applies to the problem of inferring haplotype frequencies of closely linked restriction-site polymorphisms.
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Title: Haplotype Inference with Boolean Constraint Solving: An Overview

Section 1: Introduction
Description 1: This section will provide an introduction to the haplotype inference problem, its significance in human genetics, and the computational challenges associated with its solution. It will also introduce the approach using pure parsimony and highlight the paper's objectives.

Section 2: Preliminaries
Description 2: This section will explain the basic biological concepts and terminologies relevant to haplotype inference, such as DNA, SNPs, and genotypes, ensuring that readers have the necessary background to understand the subsequent sections.

Section 3: Haplotype Inference
Description 3: This section will delve into the concept of haplotypes, the challenges in directly obtaining them from genotypes, and different computational approaches used to infer haplotypes, including the use of statistical and parsimony-based methods.

Section 4: Standard Techniques for Solving HIPP
Description 4: This section will discuss general techniques used in preprocessing and solving the HIPP problem, emphasizing methods to simplify problem instances and improve solver performance.

Section 5: Simplifying the Problem Instances
Description 5: This section will describe approaches to reducing the size of haplotype inference problem instances, such as identifying and removing redundant genotypes and simplifying symmetric sites.

Section 6: Computing Lower Bounds
Description 6: This section will explore techniques for computing lower bounds on the number of haplotypes required to explain a given set of genotypes, focusing on incompatibility graphs and other heuristic methods.

Section 7: Computing Upper Bounds
Description 7: This section will explain methods for computing upper bounds using algorithms like Clark's method and Delayed Selection, providing insights into their applications and limitations.

Section 8: Solving HIPP with ILP
Description 8: This section will review Integer Linear Programming (ILP) models for solving the HIPP problem, detailing different ILP approaches, including exponential-size and polynomial-size models.

Section 9: Solving HIPP with SAT
Description 9: This section will discuss the use of Boolean Satisfiability (SAT) solvers in addressing the HIPP problem, explaining the construction of CNF formulas and the advantages of SAT-based approaches.

Section 10: Solving HIPP with PBO
Description 10: This section will cover the use of Pseudo-Boolean Optimization (PBO) for haplotype inference, contrasting it with other methods and highlighting key modifications and performance improvements.

Section 11: Practical Experience
Description 11: This section will present empirical results and practical evaluations of different models and techniques on a set of challenging problem instances, comparing their efficiency and effectiveness.

Section 12: Research Directions
Description 12: This section will identify potential future research areas in haplotype inference, discussing the need for improved accuracy and additional criteria for selecting the most appropriate solutions.