On the complexity of several haplotyping problems

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On the complexity of several haplotyping problems

• Rudi Cilibrasi
• Steven Kelk
• John Tromp
• Leo van Iersel

Part of this research has been funded by the
Dutch BSIK/BRICKS project
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Overview

• Problem 1 Minimum Error Correction
• Problem 2 Longest Haplotype Reconstruction
• Problem 3 Pure Parsimony Haplotyping

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• DNA chromosome pairs
• Chromosome string over A,C,G,T
• SNP site where variability is observed
• Haplotype string over 0,1 representing the
  SNPs of a chromosome

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• SNP-matrix elements from 0,1,-, every row is
  a haplotype fragment, columns correspond to SNPs
• - is a hole this fragment gives no information
  at this SNP
• Gap block of holes, flanked by non-holes
• Example ---00101-----101011-----

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• Two rows are said to be in conflict if, at some
  SNP, one row has a 0 and the other row has a 1
• A SNP-matrix M is called feasible ifthere exist
  two haplotypes such that every row of M has no
  conflicts with (at least) one of these haplotypes

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Feasible SNP matrix
First haplotype
Second haplotype
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Minimum Error Correction

• Input SNP matrix M
• Output smallest number of corrections needed to
  make M feasible

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Unfeasible SNP matrix
Feasible SNP matrix
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Gapless MEC is NP-hard

• Every row is gapless
• Reduction from MAX-CUT

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v7
Edge in G
v2
Encode as
Row in M
v1
v2
v3
v4
v5
v6
v7
v8
v9
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• hi hi1 (for all odd i)
• h1i 1-h2i (for all i)
• Haplotypes correspond to a cut in G
• Example cut between vertices v1, v3, v4, v7, v9
  and v3, v5, v6, v8

First haplotype
Second haplotype
v1
v2
v3
v4
v5
v6
v7
v8
v9
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• Edge in cut V-2 corrections

Edge in cut
First haplotype
Second haplotype
v1
v2
v3
v4
v5
v6
v7
v8
v9
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• Edge not in cut V corrections

Edge not in cut
First haplotype
Second haplotype
v1
v2
v3
v4
v5
v6
v7
v8
v9
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Gapless MEC is NP-hard

• Edge in cut V-2 corrections
• Edge not in cut V corrections

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1-Gap MEC is APX-hard

• Reduction from CUBIC-MAX-CUT
• Reduction is approximation preserving
• Does not work in the gapless case

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Binary-MEC

• MEC without holes
• Thought to be NP-hard
• Based on the paper Segmentation Problemsby Jon
  Kleinberg, Christos Papadimitriou and Prabhakar
  Raghavan
• Complexity of Binary-MEC is still unknown

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Summary and open problems
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Longest Haplotype Reconstruction

• Input SNP-matrix
• Objective Remove rows from the matrix to make it
  feasible such that the sum-of-lengths of the
  induced haplotypes is maximal

First haplotype
Second haplotype
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Longest Haplotype Reconstruction
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Pure Parsimony Haplotyping

• Two haplotypes can be combined into one genotype

First haplotype
Second haplotype
Genotype
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• Genotype g can be resolved by haplotypes h1 and h2

g
h1
h2

• A 2 is called an ambiguous position

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PPH Problem

• Input set of genotypes G
• Output minimum cardinality set H of haplotypes
  such that each genotype from G can be resolved by
  two haplotypes from H

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2 Ambiguous PPH Problem

• Input set of genotypes G with at most 2
  ambiguous positions per genotype
• Output minimum cardinality of a set H of
  haplotypes such that each genotype from G can be
  resolved by two haplotypes from H

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2-amb PPH is polynomial time solvable

• Construct bipartite graph B
• Maximum Independent Set in B solves 2-amb Pure
  Parsimony Haplotyping
• MIS is polynomial for bipartite graphs
• Giuseppe Lancia, Romeo Rizzi, A Polynomial
  Solution to a Special Case of the Parsimony
  Haplotyping Problem, OR Letters (2005)

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Open problems

• Complexity of Binary MEC?
• PTAS for gapless MEC?
• Constant factor approximation for 1-gap MEC?
• Constant factor approximation for 1-gap LHR?