NPcomplete Problem 2

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NP-complete Problem 2
Lecture 25

• Prof. S M Lee
• Department of Computer Science

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Encodings An encoding "e" is a mapping from a
set S to binary strings Use encodings to map
abstract problems to concrete problems
Example - Shortest Path0101001010100101011011000
10101001010
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Hamiltonian cycle of an undirected graph G (V,
E) is a simple cycle that contains each vertex in
VA hamiltonian graph is a graph that has a
hamiltonian cycle.Let m V, it takes m!
operations to determine if G (V, E) is a
hamiltonian graph If Mary claims a graph is
hamiltonian graph and provides the vertices in
order on the hamiltonian cycle then we can verify
her claim in polynomial timeThe potential cycle
is called the certificate.
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Why Are We Studying NP-complete Problems?
R

• Because we want to know what problems can be
  solved by computers

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Polynomial Reductions

• NP-complete problems are known as decision
  problems. This means that a specific item of
  input data is accepted and depending on the
  specific problem, it is required to determine if
  the instance does have the property, then the
  answer yes is returned if not, then the answer
  no is returned

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Polynomial Reduction

• Reduction of a problem P to a problem Q problem
  qs answer for t(x) must be the same as ps
  answer for x

x
T
Algorithm for Q
T(x)
Yes or No answer
(an input for p)
An input for Q
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A Simple Reduction

• Let the problem P be given a sequence of boolean
  values, does at least one of them have the value
  true
• Let Q be given a sequence of integers, is the
  maximum of the integer positive?
• Let the transformation T be defined by
  t(x1,x2,,xn)(y1,y2,,yn) where yi1 if xitrue,
  and yi0 if xifalse
• Clearly an algorithm to solve Q, when applied to
  y1,y2,,yn, solves P for x1,x2,,xn