NP Completeness
Jie Wang
University of Massachusetts Lowell
Department of Computer Science
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Complexity Upper Bound
The study of algorithms typically follows two directions:
Seek more efficient algorithms. That is, try to reduce the complexity
upper bound for solving a computational problem.
2 Establish the fact that there are no better algorithms for solving a
given problem. That is, provide a nontrivial complexity lower bound for
solving a computational problem.
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We have seen a wide range of algorithm design principles and
techniques in the first direction, which form a toolbox for tackling
algorithmic problems.
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Complexity Lower Bound
We now deal with the second direction.
It is customary to view polynomial-time algorithms as efficient.
In practice, most problems that can be solved in polynomial time can
indeed be solved in time that is a low-degree polynomial of the input
size.
How do we know if a problem cannot be solved by a polynomial-time
algorithm?
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Turing Machines
Unlike in the first direction we use random-access machines as our
computation model for its programmability, we will use Turing machines
for the study of complexity lower bound.
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P and NP
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Polynomial-Time Reductions
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NP Completeness
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Bounded Halting (BH)
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BH Is NP-Complete
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Bounded Tiling (BT)
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BT Is NP-Complete
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Proof Continued
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Proof Continued
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Boolean Satisfiability (SAT)
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Cook’s Theorem
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Proof Continued
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Proof Continued
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Proof Continued
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3SAT
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3SAT Is NP-Complete
Theorem 7. 3SAT is NP-complete.
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Proof
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Proof Continued
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