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Problem Solving 5 Applications of NP-hardness

Problem Solving 5 Applications of NP-hardness. Knapsack. Knapsack. Knapsack is NP-hard. Decision Version of Knapsack. Partition. Set-Cover. Set-Cover.

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Problem Solving 5 Applications of NP-hardness

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  1. Problem Solving 5Applications 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. Nonunique Probe Selection and Group Testing

  14. DNA Hybridization

  15. 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)

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

  17. DNA Library Screening

  18. 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.

  19. Binary Matrix viruses c1 c2 c3 cjcn p1 0 0 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0 p2 0 1 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0 p3 1 0 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0 probes 0 0 1 … 0 … 0 … 0 … 0 … 0 … 0 … 0 . . pi 0 0 0 … 0 … 0 … 1 … 0 … 0 … 0 … 0 . . pt 0 0 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0 The cell (i, j) contains 1 iff the ith probe hybridizes the jth virus.

  20. Binary Matrix of Example virus c1 c2 c3 cj p1 1 1 1 0 0 0 0 0 0 p2 0 0 0 1 1 1 0 0 0 p3 0 0 0 0 0 0 1 1 1 probes 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 Observation: All columns are distinct. To identify up to d viruses, all unions of up to d columns should be distinct!

  21. _ d-Separable Matrix viruses c1 c2 c3 cjcn p1 0 0 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0 p2 0 1 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0 p3 1 0 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0 probes 0 0 1 … 0 … 0 … 0 … 0 … 0 … 0 … 0 . . pi 0 0 0 … 0 … 0 … 1 … 0 … 0 … 0 … 0 . . pt 0 0 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0 All unions of up to d columns are distinct. Decoding: O(nd)

  22. d-Disjunct Matrix viruses c1 c2 c3 cjcn p1 0 0 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0 p2 0 1 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0 p3 1 0 0 … 0 … 0 … 0 … 0 … 0 … 0 … 0 probes 0 0 1 … 0 … 0 … 0 … 0 … 0 … 0 … 0 . . pi 0 0 0 … 0 … 0 … 1 … 0 … 0 … 0 … 0 . . pt 0 0 0 … 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.

  23. 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

  24. Problem

  25. 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). -

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

  27. - 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. -

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

  29. - 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. -

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

  31. 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 NP-hard. • In general, the problem is conjectured to be Σ2 –complete. p

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