1.2 Genome-Wide Association Studies . . . . 2

C O N T E N T S

1 introduction 1

1.1 Motivation . . . . 1

1.2 Genome-Wide Association Studies . . . . 2

1.3 Combinatorial Optimization . . . . 4

1.4 Outline of the Thesis . . . . 11

2 a class representative model for pph- ud 15 2.1 Notation And Problem Statement . . . . 20

2.2 Design and Implementation . . . . 26

2.2.1 Reducing model size . . . . 31

2.2.2 Accounting for perfect phylogeny constraints . 33 2.3 Experimental results . . . . 35

2.3.1 Implementation . . . . 36

2.3.2 Benchmark instances . . . . 36

2.3.3 Computational performances . . . . 37

3 a branch - and-price algorithm for the plohp 47 3.1 Notation and Problem Formulation . . . . 50

3.2 A Branch-and-Price algorithm for the PLOHP . . . . . 55

3.2.1 An integer linear programming formulation for the Parsimonious Loss of Heterozygosity Problem (

) . . . . 55

3.2.2 Finding a maximum node-weighted clique in a Max-Point Tolerance Graph (

) . . . . 58

3.2.3 Initializing the columns of the RMP . . . . 60

3.2.4 Branching rules . . . . 61

3.2.5 Preprocessing . . . . 61

3.3 Experimental Results . . . . 64

3.3.1 Implementation . . . . 65

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3.3.2 Calibrating the pricing oracle . . . . 65

3.3.3 Benchmark instances . . . . 68

3.3.4 Computational performances . . . . 71

3.3.5 Computational performances on larger datasets of the

. . . . 72

4 an ip formulation of the mipairp 77 4.1 Notation and Problem Statement . . . . 79

4.2 The ALU Graph . . . . 83

4.3 The complexity of PAIRP and MIPAIRP . . . . 85

4.4 An Integer Programming Model for the MIPAIRP . . . 92

4.5 Experimental Results . . . . 94

4.5.1 Implementation . . . . 95

4.5.2 Benchmark Instances . . . . 95

4.5.3 Computational performances . . . . 96

5 conclusion 99 5.1 Contribution . . . . 99

5.2 Perspectives . . . . 101

bibliography 105