Complexity and Approximation of the Minimum Recombinant Haplotype Configuration Problem

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Complexity and Approximation of the Minimum
Recombinant Haplotype Configuration Problem
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Outline

• Introduction and problem definition
• Deciding the complexity of binary-tree-MRHC
• Approximation of MRHC with missing data
• Approximation of MRHC without missing data
• Approximation of bounded MRHC
• Conclusion

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Introduction

• Basic concepts

• Mendelian Law one haplotype comes from the
  mother and the other comes from the father.

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Notations and Recombinant
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Pedigree

• An example British Royal Family

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Haplotype Reconstruction

• - Haplotype useful, expensive
• - Genotype cheaper

• Reconstruct haplotypes from genotypes

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Problem Definition

• MRHC problem
• Given a pedigree and the genotype
  information for each member, find a haplotype
  configuration for each member which obeys
  Mendelian law, s.t. the number of recombinants
  are minimized.

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Problem Definition

• Variants of MRHC
• Tree-MRHC no mating loop
• Binary-tree-MRHC 1 mate, 1 child
• 2-locus-MRHC 2 loci
• 2-locus-MRHC 2 loci with missing data

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Previous Work
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Our hardness and approximation results
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Our hardness and approximation results
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Outline

• Introduction and problem definition
• Deciding the complexity of binary-tree-MRHC
• Approximation of MRHC with missing data
• Approximation of MRHC without missing data
• Approximation of bounded MRHC
• Conclusion

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A verifier for ?3SAT (1)

• Given a truth assignment for literals in a
  3CNF formula
• Consistency checking for each variable
• Satisfiability checking for each clause

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Binary-tree-MRHC is NP-hard
C can check if M have certain haplotype
configuration!!
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Binary-tree-MRHC is NP-hard
?3SAT is satisfiable ? OPT(MRHC)clauses
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Outline

• Introduction and problem definition
• Deciding the complexity of binary-tree-MRHC
• Approximation of MRHC with missing data
• Approximation of MRHC without missing data
• Approximation of bounded MRHC
• Conclusion

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Inapproximability of 2-locus -MRHC

• Definition A minimization problem R cannot be
  approximated

-There is not an approximation algorithm with
ratio f(n) unless PNP. -f(n) is any
polynomial-time computable function

• Fact If it is NP-hard to decide whether
  OPT(R)0, R cannot be approximated unless PNP.

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Inapproximability of 2-locus -MRHC

• Reduce 3SAT to 2-locus-MRHC

2-locus-MRHC cannot be approximatedunless
PNP!!

• 3SAT is satisfiable?OPT(2-locus-MRHC)0

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Outline

• Introduction and problem definition
• Deciding the complexity of binary-tree-MRHC
• Approximation of MRHC with missing data
• Approximation of MRHC without missing data
• Approximation of bounded MRHC
• Conclusion

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Upper Bound of 2-locus-MRHC

• Main idea use a Boolean variable to capture the
  configuration
• use clauses to capture the
  recombinants.

• An example

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Upper Bound of 2-locus-MRHC

• The reduction from 2-locus-MRHC to Min 2CNF
  Deletion

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Upper Bound of 2-locus-MRHC

• Recently, Agarwal et al. STOC05 presented an
• O ( ) randomized approximation
  algorithm
• for Min 2CNF Deletion.

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Outline

• Introduction and problem definition
• Deciding the complexity of binary-tree-MRHC
• Approximation of MRHC with missing data
• Approximation of MRHC without missing data
• Approximation of bounded MRHC
• Conclusion

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Approximation Hardness of bounded MRHC

• Bound mates and children
• 2-locus-MRHC (16,15)
• 2-locus-MRHC (4,1)
• tree-MRHC (u,1) or (1,u)

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Conclusion

• Our hardness and approximation results

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• Thanks for your time
• and attention!

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L-Reduction

• Given two NPO W, Q and a polynomial time
  transformation p from instances of W to instances
  of Q, p is an Lreduction if PY91
• OPTQ(p(? )) a OPTw (? ), and
• For every feasible solution q of p(? ) with
  objective value SOLQ(p(? ), q), we can find in
  polynomial time a solution g(q) to ? with
  objective value SOLW(? , g(q)) such that OPTW
  (? )- SOLW(? , g(q)) ß OPTQ(p(? ))- SOLQ(p(?
  ), q).

If W cant be approximated within ratio r
unless PNP, Q cant be approximated within ratio
unless PNP !!
r (a ß 1)
(r-1)/(aß)1