Lecture 1011 Polynomial Time Hierarchy

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Lecture 10-11Polynomial Time Hierarchy
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Nonunique Probe Selection and Group Testing
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DNA Hybridization
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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(lt5kb)

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Preparation of a DNA Library

• DNA library a collection of cloned DNA fragments
• usually from a specific organism

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DNA Library Screening
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Problem

• If a probe doesnt 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.

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Binary Matrix

• viruses
• c1 c2 c3 cj cn
• 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.

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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!

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d-Separable Matrix
_

• viruses
• c1 c2 c3 cj cn
• 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)

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d-Disjunct Matrix

• viruses
• c1 c2 c3 cj cn
• 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.

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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

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Problem
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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).

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d-Separability Test

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

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d-Separability Test
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-

• 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.

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(b) Reduction from vertex-cover

• Given a graph G and h gt 0, does G have a vertex
  cover of size at most h?

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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.

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d-Disjunct Test

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

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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 gt0, the problem is NP-hard.
• In general, the problem is conjectured to be S2
  complete.

p
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Oracle TM
Query tape
yes
Query state
no
Remark
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A lt T B
p

• A is polynomial-time Turing-reducible to B if
  there exists a polynomial-time DTM with oracle B,
  accepting A.
• A e P
• A lt M B gt A lt T B

B
p
p
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Example
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Definition
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Definition
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Definition
Remark
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Theorem
Proof
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PSPACE
P
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Characterization
THEOREM
PROOF by induction on k
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Induction Step on kgt1
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Characterization
THEOREM
PROOF by induction on k
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Induction Step on kgt1