1. Lecture 2-6 Polynomial Time Hierarchy

2. Nonunique Probe Selection and Group Testing

3. DNA Hybridization

4. Polymerase Chain Reaction (PCR)
cell-free method of DNA cloning
Advantages
much faster than cell based method
need very small amount of target DNA
Disadvantages
need to synthesize primers
applies only to short DNA fragments(<5kb)

5. Preparation of a DNA Library
DNA library: a collection of cloned DNA fragments
usually from a specific organism

6. DNA Library Screening

7. Problem
If a probe doesn’t uniquely determine a virus, i.e., a probe determine a group of viruses, how to select a subset of probes from a given set of probes, in order to be able to find up to d viruses in a blood sample.

8. Binary Matrix viruses c1 c2 c3 cj cn
p … 0 … 0 … 0 … 0 … 0 … 0 … p … 0 … 0 … 0 … 0 … 0 … 0 … 0
p … 0 … 0 … 0 … 0 … 0 … 0 … 0
probes … 0 … 0 … 0 … 0 … 0 … 0 … 0 .
pi … 0 … 0 … 1 … 0 … 0 … 0 … 0
pt … 0 … 0 … 0 … 0 … 0 … 0 … 0
The cell (i, j) contains 1 iff the ith probe hybridizes the jth virus.

9. Binary Matrix of Example
virus
c1 c2 c cj
p p p
probes
Observation: All columns are distinct.
To identify up to d viruses, all unions of up to d columns should be distinct!

10. d-Separable Matrix _ All unions of up to d columns are distinct.
viruses
c1 c2 c cj cn
p … 0 … 0 … 0 … 0 … 0 … 0 … 0
p … 0 … 0 … 0 … 0 … 0 … 0 … 0
p … 0 … 0 … 0 … 0 … 0 … 0 … 0
probes … 0 … 0 … 0 … 0 … 0 … 0 … 0 .
pi … 0 … 0 … 1 … 0 … 0 … 0 … 0
pt … 0 … 0 … 0 … 0 … 0 … 0 … 0
All unions of up to d columns are distinct.
Decoding: O(nd)

12. Nonunique Probe Selection_
Given a binary matrix, find a d-separable submatrix with the same number of columns and the minimum number of rows.
Given a binary matrix, find a d-disjunct submatrix with the same number of columns and the minimum number of rows.
Given a binary matrix, find a d-separable submatrix with the same number of columns and the minimum number of rows

13. Complexity? All three problems may not be in NP, why?
Guess a t x n matrix M, verify if M is d-separable (d-separable, d-disjunct). -

14. Problem

15. - d-Separability Test - Given a matrix M and d, is M d-separable?
It is co-NP-complete.

16. - d-Separability Test - Given a matrix M and d, is M d-separable?
This is co-NP-complete.
(a) It is in co-NP.
Guess two samples from space S(n,d). Check if M gives the same test outcome on the two samples. -

17. (b) Reduction from vertex-cover
Given a graph G and h > 0, does G have a vertex cover of size at most h?

19. d-Separability Test Reduces to d-Separability Test
Put a zero column to M to form a new matrix M*
Then M is d-separable if and only if M* is d-separable. -

20. d-Disjunct Test Given a matrix M and d, is M d-disjunct?
This is co-NP-complete.

21. Minimum d-Separable Submatrix
Given a binary matrix, find a d-separable submatrix with minimum number of rows and the same number of columns.
For any fixed d >0, the problem is co-NP-hard.
In general, the problem is in and conjectured to be –complete.

23. Polynomial Time Hierarchy

24. PSPACE PH P

25. Oracle TM
Query tape
yes
Query
state no
Remark:

26. A is polynomial-time Turing-reducible to B if there exists a polynomial-time DTM with oracle B, accepting A.

27. Example

28. Example

29. Definition

30. Definition

31. Definition
Remark

32. Theorem
Proof

33. PSPACE PH P

34. Characterization
THEOREM
PROOF by induction on k

35. Induction Step on k>1

40. Characterization
THEOREM
PROOF by induction on k

41. Induction Step on k>1