Complexity and Approximation of the Minimum Recombinant Haplotype Configuration Problem - PowerPoint PPT Presentation

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Complexity and Approximation of the Minimum Recombinant Haplotype Configuration Problem

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Complexity and Approximation of the Minimum Recombinant Haplotype Configuration Problem Authors: Lan Liu, Xi Chen, Jing Xiao & Tao Jiang – PowerPoint PPT presentation

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Title: Complexity and Approximation of the Minimum Recombinant Haplotype Configuration Problem


1
Complexity and Approximation of the Minimum
Recombinant Haplotype Configuration Problem
2
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

3
Introduction
  • Basic concepts
  • Mendelian Law one haplotype comes from the
    mother and the other comes from the father.

4
Notations and Recombinant
5
Pedigree
  • An example British Royal Family

6
Haplotype Reconstruction
  • - Haplotype useful, expensive
  • - Genotype cheaper
  • Reconstruct haplotypes from genotypes

7
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.

8
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

9
Previous Work
10
Our hardness and approximation results
11
Our hardness and approximation results
12
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

13
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

14
Binary-tree-MRHC is NP-hard
C can check if M have certain haplotype
configuration!!
15
Binary-tree-MRHC is NP-hard
?3SAT is satisfiable ? OPT(MRHC)clauses
16
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

17
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.

18
Inapproximability of 2-locus -MRHC
  • Reduce 3SAT to 2-locus-MRHC

2-locus-MRHC cannot be approximated unless
PNP!!
  • 3SAT is satisfiable?OPT(2-locus-MRHC)0

19
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

20
Upper Bound of 2-locus-MRHC
  • Main idea use a Boolean variable to capture the
    configuration
  • use clauses to capture the
    recombinants.
  • An example

21
Upper Bound of 2-locus-MRHC
  • The reduction from 2-locus-MRHC to Min 2CNF
    Deletion

22
Upper Bound of 2-locus-MRHC
  • Recently, Agarwal et al. STOC05 presented an
  • O ( ) randomized approximation
    algorithm
  • for Min 2CNF Deletion.

23
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

24
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)

25
Conclusion
  • Our hardness and approximation results

26
  • Thanks for your time
  • and attention!

27
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
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