1. NP-Completeness A problem is NP-complete if: It is in NP
Every NP problem is reduced to it
(in polynomial time)

2. Observation:
If we can solve any NP-complete problem
in Deterministic Polynomial Time (P time)
then we know:

3. Observation:
If we prove that
we cannot solve an NP-complete problem
in Deterministic Polynomial Time (P time)
then we know:

4. Cook’s Theorem:
The satisfiability problem is NP-complete
Proof:
Convert a Non-Deterministic Turing Machine
to a Boolean expression
in conjunctive normal form

5. An NP-complete Language
Cook-Levin Theorem:
Language SAT (satisfiability problem)
is NP-complete
Proof:
Part1:
SAT is in NP
(we have proven this in previous class)
Part2: reduce all NP languages
to the SAT problem
in polynomial time
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6. Other NP-Complete Problems:
The Traveling Salesperson Problem
Vertex cover
Hamiltonian Path
All the above are reduced
to the satisfiability problem

7. Observations:
It is unlikely that NP-complete
problems are in P
The NP-complete problems have
exponential time algorithms
Approximations of these problems
are in P

8. NP-complete Languages
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9. Polynomial Time Reductions
Polynomial Computable function :
There is a deterministic Turing machine
such that for any string computes
in polynomial time:
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10. is polynomial time reducible to language
Definition:
Language
is polynomial time reducible to
language
if there is a polynomial computable
function such that:
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11. Suppose that is polynomial reducible to . If then .
Theorem:
Suppose that is polynomial reducible to .
If then
Proof:
Let be the machine that decides
in polynomial time
Machine to decide in polynomial time:
On input string :
1. Compute
2. Run on input
3. If acccept
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12. Example of a polynomial-time reduction:
We will reduce the
3CNF-satisfiability problem
to the
CLIQUE problem
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13. Each clause has three literals
3CNF formula:
literal
clause
Each clause has three literals
Language:
3CNF-SAT ={ : is a satisfiable
3CNF formula}
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14. A 5-clique in graph Language: CLIQUE = { : graph contains a -clique}
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15. 3CNF-SAT is polynomial time reducible to CLIQUE
Theorem:
3CNF-SAT is polynomial time
reducible to CLIQUE
Proof:
give a polynomial time reduction
of one problem to the other
Transform formula to graph
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16. Transform formula to graph. Example:
Clause 2
Create Nodes:
Clause 3
Clause 1
Clause 4
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17. Add link from a literal to a literal in every
other clause, except the complement
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18. Resulting Graph
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19. End of Proof The formula is satisfied if and only if
the Graph has a 4-clique
End of Proof
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20. NP-complete Languages
We define the class of NP-complete
languages
Decidable NP
NP-complete
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21. A language is NP-complete if:
is in NP, and
Every language in NP
is reduced to
in polynomial time
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22. If a NP-complete language is proven to be in P then:
Observation:
If a NP-complete language
is proven to be in P then:
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23. Decidable NP P
NP-complete ?
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