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

3. 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

4. Introduction Basic concepts Mendelian Law: one haplotype comes from the mother and the other comes from the father. Example: Mendelian experiment

5. Notations and Recombinant 11111111 22222222 22222222 22222222 11111111 0 recombinant 22222222 Father Mother : recombinant 11111111 22222222 22222222 22222222 11221122 22222222 1 recombinant Father Mother 11221122 22222222 Genotype 12221222 21222122 Haplotype Configuration

6. Pedigree An example: British Royal Family

7. Haplotype Reconstruction - Haplotype: useful, expensive - Genotype: cheaper Reconstruct haplotypes from genotypes

8. 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.

9. 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

10. Previous Work  The known hardness results for Mendelian law checking  The known hardness results for MRHC

11. Our hardness and approximation results

13. 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

14. 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

15. Binary-tree-MRHC is NP-hard (A) C’s genotype (B) Two haplotype configurations C can check if M have certain haplotype configuration!!

16. Binary-tree-MRHC is NP-hard The pedigree ≠3SAT is satisfiable  OPT(MRHC)=#clauses

17. 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

18. Inapproximability of 2-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.

19. Inapproximability of 2-locus -MRHC* Reduce 3SAT to 2-locus-MRHC* 3SAT is satisfiable  OPT(2-locus-MRHC*)=0 2-locus-MRHC* cannot be approximated unless P=NP!! False True

20. 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

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

22. Upper Bound of 2-locus-MRHC The reduction from 2-locus-MRHC to Min 2CNF Deletion

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

24. 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

25. 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)

26. Conclusion  Our hardness and approximation results

27. Thanks for your time and attention!

28. L-Reduction Given two NPO W, Q and a polynomial time transformation π from instances of W to instances of Q, π is an L­reduction if [PY91]  OPT Q (π( ω )) ≤ α· OPT w ( ω ), and  For every feasible solution q of π( ω ) with objective value SOL Q (π( ω ), q), we can find in polynomial time a solution g(q) to ω with objective value SOL W ( ω, g(q)) such that | OPT W ( ω )- SOL W ( ω, g(q))| ≤ β| OPT Q (π( ω ))- SOL Q (π( ω ), q)|. If W can’t be approximated within ratio r unless P=NP, Q can’t be approximated within ratio unless P=NP !! r ( α = β =1 ) (r-1)/( α β)+1