Complete Parsimony Haplotype Inference Problem and Algorithms

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.

Haplotype inference by pure parsimony (HIPP) is known to be NP-Hard. Despite this, many algorithms successfully solve HIPP instances on simulated and real data. In this paper, we explore the connection between algebraic rank and the HIPP problem, to help identify easy and hard instances of the problem. The rank of the input matrix is known to be a lower bound on the size an optimal HIPP solution. We show that this bound is almost surely tight for data generated by randomly pairing p haplotypes derived from a perfect phylogeny when the number of distinct population members is more than (1 + epsilon/2)p log p (for some positive epsilon). Moreover, with only a constant multiple more population members, and a common mutation, we can almost surely recover an optimal set of haplotypes in polynomial time. We examine the algebraic effect of allowing recombination, and bound the effect recombination has on rank. In the process, we prove a stronger version of the standard haplotype lower bound. We also give a complete classification of the rank of a haplotype matrix derived from a galled tree. This classification identifies a set of problem instances with recombination when the rank lower bound is also tight for the HIPP problem.

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.

Haplotype inference is a challenging problem in bioinformatics that consists in inferring the basic genetic constitution of diploid organisms on the basis of their genotype. This piece of information allows researchers to perform association studies for the genetic variants involved in diseases and the individual responses to therapeutic agents. A notable approach to the problem is to encode it as a combinatorial problem (under certain hypotheses, such as the pure parsimony) and to solve it using off-the-shelf combinatorial optimization techniques. In this paper, we present and discuss an approach based on hybridization of two meta-heuristics, one being a population based learning algorithm and the other a local search. We test our approach by solving instances from common Haplotype inference benchmarks. Results show that this approach achieves an improvement on solution quality with respect to the application of a single 'pure' algorithm.

The haplotype resolution from xor-genotype data has been recently formulated as a new model for genetic studies [1]. The xor-genotype data is a cheaply obtainable type of data distinguishing heterozygous from homozygous sites without identifying the homozygous alleles. In this paper we propose a formulation based on a well known model used in haplotype inference: pure parsimony. We exhibit exact solutions of the problem by providing polynomial-time algorithms for some restricted cases and a fixed-parameter algorithm for the general case. These results are based on some interesting combinatorial properties of a graph representation of the solutions. Moreover we propose a heuristic and produce an experimental analysis showing that it scales to real-world instances taken from the HapMap project.

Haplotype inference is a challenging computational problem in population genetics. We introduce an approach using Ant Colony Optimization (ACO) metaheuristic, named ACOHAP, to infer haplotypes from unphased Single Polymorphism Nucleotide (SNP) marker data. Our method employs an efficient method for constructing the ACO graph through which ants flexibly traverse to build haplotypes. ACOHAP also uses a well-performed pheromone trail update strategy and a local search to improve the performance. Experiments showed that ACOHAP outperformed the state-of-the-art methods for haplotype inference in both simulated and biological data.

To analyze the function of DNA, researchers have to obtain each haplotype, the genetic constitution of an individual chromosome, of an individual for analysis. Due to the significant efforts required in collecting haplotypes, the descriptions of one conflated pair of haplotypes called genotypes are usually collected. Since the genotype data contains insufficient information to identify the combination of DNA sequence in each copy of a chromosome, one has to solve the population haplotype inference problem by pure parsimony criterion which uses the minimum number of haplotypes to infer the haplotype data from genotype data for a population. Previous researches use mathematical programming methods such as integer programming and semidefinite programming models to solve the population haplotype inference problem. However, no computational experiment has ever been conducted to evaluate the algorithmic effectiveness. This paper thus conducts the first computational experiments on four haplotyping algorithms, including our new greedy heuristic and three pervious haplotyping algorithms.

Solution method for the location planning problem of logistics park with variable capacity

A Survey of Lagrangean Techniques for Discrete Optimization

This paper describes the details of a successful application where an integer programming and evolutionary hybrid algorithm was used to solve a bus driver duty optimization problem. The task is NP-hard, therefore theoretically optimal solutions can only be calculated for very small problem instances. Our aim is to obtain solutions of good quality within reasonable time limits. We first applied an integer programming approach to a set partitioning problem. The model was solved with a column generation algorithm in a branch and bound scheme. In order to solve larger real-life problems, we have combined the integer programming method with a greedy 1+1 steady state evolutionary algorithm. The resulting hybrid algorithm was capable of providing near-optimal solutions within reasonable timescales to larger instances of the bus driver scheduling problem. We present the results and running times of our algorithm in detail, as well as possible directions of future improvements.

Ranking in quadratic integer programming problems

Mathematical properties and bounds on haplotyping populations by pure parsimony

Integer Linear Programming (ILP) is commonly used in high level and system level synthesis. It is an NP-Complete problem (in general cases). There exists some tools that give an optimal solution for small ILP formulations. Nevertheless, these tools may not give solutions for complex formulations. In this paper, we present a solution to overcome the problem of complexity in ILP formulations. We propose a partitioning methodology based on a constraint graph representing all the constraints included in any ILP formulation. To direct the partitioning, the constraint graph nodes are grouped to represent Data Flow Graph (DFG) nodes. This reduced constraint graph can be used to partition any ILP formulation based on DFG. We illustrate this method on ILP formulation for scheduling. Results on ILP scheduling formulations are promising.

Exact solution approaches for integer linear generalized maximum multiplicative programs through the lens of multi-objective optimization

Integer Linear Programming (ILP) can be seen as the archetypical problem for NP-complete optimization problems, and a wide range of problems in artificial intelligence are solved in practice via a translation to ILP. Despite its huge range of applications, only few tractable fragments of ILP are known, probably the most prominent of which is based on the notion of total unimodularity. Using entirely different techniques, we identify new tractable fragments of ILP by studying structural parameterizations of the constraint matrix within the framework of parameterized complexity. In particular, we show that ILP is fixed-parameter tractable when parameterized by the treedepth of the constraint matrix and the maximum absolute value of any coefficient occurring in the ILP instance. Together with matching hardness results for the more general parameter treewidth, we draw a detailed complexity landscape of ILP w.r.t. decompositional parameters defined on the constraint matrix.