1. Lecture 2-5 Applications of NP-hardness

2. Knapsack

3. Knapsack

4. Knapsack is NP-hard

5. Decision Version of Knapsack

6. Partition

7. Set-Cover

8. Set-Cover
Given a collection C of subsets of a set X, find a minimum subcollection C’ of C such that every element of X appears in a subset in C’ . (Such a C’ is called a set-cover.)

9. Vertex-Cover

10. Decision Version of Vertex-Cover
Decision Version of Set-Cover

11. Broadcast in Multi-Channel Wireless Networks

12. Problem

13. Hitting Set
Given a collection C of subsets of a set X, find a minimum subset Y of X such that intersection of Y and S is not empty for every S in C.

14. Nonunique Probe Selection and Group Testing

15. DNA Hybridization

16. 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)

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

18. DNA Library Screening

19. 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.

20. 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.

21. 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!

22. 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)

24. d-Disjunct 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
Each column is different from the union of every d other columns
Decoding: O(n)
Remove all clones in negative pools. Remaining clones are all
positive.

25. 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

26. 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). -

27. Problem

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

29. - 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. -

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

32. 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. -

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

34. 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 conjectured to be Σ2 –complete. p