ADVANCED COMPUTATIONAL MODELS AND ALGORITHMS

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ADVANCED COMPUTATIONAL MODELS AND ALGORITHMS

• Instructor Dr. GautamDas
• February 24, 2009
• Class notes by
• Ranganath M R

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Overview

• NP Hard Problems
• P Space (Polynomial Space)
• Quantified SAT (P Space complete problem)
• Deterministic Approximation Algorithms
• Vertex Cover approximation algorithm.

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NP Hard Problems

• A problem p belongs to NP Hard if,
• p is as hard as any NP complete problem i.e. if
  any NP complete problem p p
• ii is not known to have a polynomial
  verification time for p.

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P Space (Polynomial space)

• A problem p belongs to P Space if,
• There exists an algorithm to solve the
  problem that only requires a polynomial
  amount of extra memory.
• Input n
• Extra memory polynomial (n)
• The P Space extends even outside the NP complete
  space and Co-NP Space

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Example

• The SAT problem is closed under P Space.
• We can have an integer whose length in bits is
  equal to the number of variables of the SAT
  problem. Example if we 8 variables in SAT, then
  we take an integer with 8 bits(byte). Each bit of
  the integer represents the variable of the SAT
  problem. Hence the entire truth assignments can
  be verified in extra space which is poynomial.

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Quantified SAT problem

• Example let f (X1 v X2) (X1 v X3bar )
• (X2 v X1 v X3bar v X4 bar )
• Does there exist X1 and X2 such that for all X3
  and X4
• f(X1, X2,X3,X4) 1 (true).
• This Problem is P space complete as this problem
  takes extra memory which is not polynomial. This
  is because we should check the function F, for
  every value of X3 and X4 when X1 and X2 are
  fixed. Hence this problem is P Space complete.

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• In general, if a problem of the sort
• ( there exists .For all .. There
  existsfor all) are all P space complete.

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Deterministic Approximation Algorithms

• Vertex Cover Problem
• Input G(V,E)
• Vertex Cover Problem Approx algorithm A for
  the VC problem is one that produces a VC ,
  Vapprox such that
• A runs in poly time
• 1 (Vapprox / Vopt ) C.
• Vopt is the optimal Vertex cover for the same
  graph
• C is some constant.

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An approximation algorithm for VC

• Lets us consider this graph for the VC problem.

A
B
E
C
D
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Approximation Algorithm for calculating VC

• Step1 Select any edge, then add the two nodes
  which are at the edge into the VC set.
• Step2 Delete all the edges originating from
  those two verticies.
• Step3 If all the edges are not covered, then go
  to step 1
• Else we have the VC being the approximate
  vertex cover set.

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Example of the problem

• Select edge BE. Vapprox B,E

A
B
E
C
D
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• Now the edges are originating from B and E are
  removed . VC B,E

A
B
E
C
D
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• Now select AC. Vapprox B,E,A,C

A
B
E
C
D
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• Now no edges are remaining hence Vapprox
  B,E,A,C

A
B
E
C
D
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• Now, the Vopt , should atleast contain two of
  the vertices or more.

A
B
E
C
D
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• Hence in general, the if the Vapprox contains 2L
  nodes, the Vopt will contain atleast L nodes.
  Hence C in this case is less than or equal to
  2L/L.