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

Transcription

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

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. Genotype Locus Haplotype PS value= PS value=0 Example: Mendelian experiment

4 Notations and Recombinant 0 recombinant Father Mother : recombinant recombinant Father Mother Genotype Haplotype Configuration

5 Pedigree An example: British Royal Family Elizabeth II of the United Kingdom Prince Philip, Duke of Edinburgh Diana, Princess of Wales Prince Charles, Prince of Wales Camilla, Duchess of Cornwall Captain Mark Phillips Princess Anne, Princess Royal Commander Timothy Laurence Prince Andrew, Duke of York Sarah Margaret Ferguson Prince Edward, Earl of Wessex Sophie Rhys-Jones Prince William of Wales Prince Henry of Wales Peter Phillips Zara Phillips Princess Beatrice of York Princess Eugenie of York Lady Louise Windsor

6 Haplotype Reconstruction - Haplotype: useful, expensive - Genotype: cheaper Reconstruct haplotypes from genotypes M C M C M C (a) (b)

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: mate, child -locus-mrhc: loci -locus-mrhc*: loci with missing data

9 Previous Work The known hardness results for Mendelian law checking Loop? Multiallelic? Hardness Yes Yes NP-hard [AHI+03] No P [AHI+03] No P [AHI+03] The known hardness results for MRHC -locus-mrhc Tree-MRHC with bounded #members Tree-MRHC with bounded #loci Tree-MRHC Hardness NP-hard [LJ03] P [LJ03] P [DLJ03] NP-hard [DLJ03]

10 Our hardness and approximation results Binary-tree- MRHC -locus-mrhc* Binary-tree- MRHC* -locus-mrhc Tree-MRHC Hardness NP Lower bound of approx. ratio Any f(n) Any f(n) Any constant Any constant Assumption P NP P NP P NP the Unique Games Conjecture[Khot0] P NP the Unique Games Conjecture The lower bound holds for -locus-mrhc* (4,) Binary-tree- MRHC*(,) -locus-mrhc (6,5) Tree-MRHC(,u) Tree-MRHC(u,) Upper bound of approx. ratio O ( ) log(n)

11 Our hardness and approximation results Binary-tree- MRHC -locus-mrhc* Binary-tree- MRHC* -locus-mrhc Tree-MRHC Hardness NP Lower bound of approx. ratio Any f(n) Any f(n) Any constant Any constant Assumption P NP P NP P NP the Unique Games Conjecture[Khot0] P NP the Unique Games Conjecture The lower bound holds for -locus-mrhc* (4,) Binary-tree- MRHC*(,) -locus-mrhc (6,5) Tree-MRHC(,u) Tree-MRHC(u,) Upper bound of approx. ratio O ( ) log(n)

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 () 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 (A) C s genotype (B) Two haplotype configurations M C M M C (a) (b) (c) C C can check if M have certain haplotype configuration!!

15 Binary-tree-MRHC is NP-hard O O A B M C A B M... C A t B t 3SAT is satisfiable OPT(MRHC)=#clauses consistency checking M t- C t A t+ B t+ M t C t+ A t+ B t+ Part (#recombinants >=0 ) M t+ C t+ A t+ 3 B t+3 Part (#recombinants >=#clauses) sa tisfiab ility checking M t+ C t+ 3 M t+ 3m -... A t+ 3m C t+3m B t+ 3m The pedigree M t+ 3m

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 -locus -MRHC* Definition: A minimization problem R cannot be approximated -There is not an approximation algorithm with ratio f(n) unless P=NP. -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 P=NP.

18 Inapproximability of -locus -MRHC* Reduce 3SAT to -locus-mrhc* x True False (A) gadget for variable x x x * * y * * (B) gadget for clause x z y z * -locus-mrhc* cannot be approximated unless P=NP!! 3SAT is satisfiable OPT(-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 -locus-mrhc Main idea: use a Boolean variable to capture the configuration; use clauses to capture the recombinants. True False An example A B A B ( A B )

21 Upper Bound of -locus-mrhc The reduction from -locus-mrhc to Min CNF Deletion Genotype of the Mother (A) Genotype of the Father (B) X X Y X X X Y X Y X X X Y X X X Genotype of the Child (C) CNF Constraint ( A B) ( A B) ( A B) ( A B) ( A B) ( A B) ( A B) ( A B) ( A C) ( A C) ( B C) ( B C) Y X X X Y X Y Y X X X Y Y Y X Y A A ( A C) ( A C) A A A A

22 Upper Bound of -locus-mrhc Recently, Agarwal et al. [STOC05] presented an O ( log(n) ) randomized approximation algorithm for Min CNF Deletion. -locus-mrhc has O ( log(n) ) approximation algorithm.

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 -locus-mrhc: (6,5) -locus-mrhc*: (4,) tree-mrhc: (u,) or (,u)

25 Conclusion Our hardness and approximation results Binary-tree- MRHC -locus-mrhc* Binary-tree- MRHC* -locus-mrhc Tree-MRHC Hardness NP-hard Lower bound of approx. ratio Any f(n) Any f(n) Any constant Any constant Assumption P NP P NP P NP the Unique Games Conjecture P NP the Unique Games Conjecture The lower bound holds for -locus-mrhc* (4,) Binary-tree- MRHC*(,) -locus-mrhc (6,5) Tree-MRHC(,u) Tree-MRHC(u,) Upper bound of approx. ratio O ( ) log(n)

26 Thanks for your time and attention!