Lecture 10 Applications of NP-hardness. Knapsack.

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Presentation transcript:

Lecture 10 Applications of NP-hardness

Knapsack

Knapsack is NP-hard

Decision Version of Knapsack

Partition

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.)

Vertex-Cover

Decision Version of Vertex-Cover Decision Version of Set-Cover

Broadcast in Multi-Channel Wireless Networks

Problem

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.

Nonunique Probe Selection and Group Testing

DNA Hybridization

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)

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

DNA Library Screening

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.

Binary Matrix viruses c 1 c 2 c 3 c j c n 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. p i … 0 … 0 … 1 … 0 … 0 … 0 … 0. p t … 0 … 0 … 0 … 0 … 0 … 0 … 0 The cell (i, j) contains 1 iff the ith probe hybridizes the jth virus.

Binary Matrix of Example virus c 1 c 2 c 3 c j 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!

d-Separable Matrix viruses c 1 c 2 c 3 c j c n 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. p i … 0 … 0 … 1 … 0 … 0 … 0 … 0. p t … 0 … 0 … 0 … 0 … 0 … 0 … 0 All unions of up to d columns are distinct. Decoding: O(n d ) _

d-Disjunct Matrix viruses c 1 c 2 c 3 c j c n 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. p i … 0 … 0 … 1 … 0 … 0 … 0 … 0. p t … 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.

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 _

Problem

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). -

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

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

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

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. - -

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

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 Σ 2 –complete. p