1. Lecture 24 NP-Complete Problems

2. (1) Polynomial-time many-one reduction

3. A < m B p
A set A in Σ* is said to be polynomial-time many-one reducible to B in Γ* if there exists a polynomial-time computable function f: Σ* → Γ* such that
x ε A iff f(x) ε B. Σ* Γ*

4. A = Hamiltonian cycle (HC)
Given a graph G, does G contain a Hamiltonian cycle?

5. B = decision version of Traveling Salesman Problem (TSP)
Given n cities and a distance table between these n cities, find a tour (starting from a city and come back to start point passing through each city exactly once) with minimum total length.
Given n cities, a distance table and k > 0,
does there exist a tour with total length < k?

6. HC < m TSP p
From a given graph G, we need to construct (n cities, a distance table, k).

7. 3-SAT < m SAT p
SAT: Given a Boolean formula F, does F have a satisfied assignment?
An assignment is satisfied if it makes F =1.
3-SAT: Given a 3-CNF F, does F have a satisfied assignment?

8. Boolean Algebra

9. Boolean Algebra

10. 3CNF

11. Examples

13. SAT < m 3-SAT p
SAT: Given a Boolean formula F, does F have a satisfied assignment?
An assignment is satisfied if it makes F =1.
3-SAT: Given a 3-CNF F, does F have a satisfied assignment?

17. Property of < m A < m B and B < m C imply A < m C
A < m B and B ε P imply A ε P p p p

18. NP-complete A set A is NP-hard if for any B in NP, B < m A.
A set A is NP-complete if it is in NP and NP-hard.
A decision problem is NP-complete if its corresponding language is NP-complete.
An optimization problem is NP-hard if its decision version is NP-hard. p

19. (2) Cook Theorem
SAT is NP-complete

20. Proof of Cook Theorem

24. The 1st tape should be

25. The last tape should contain
The final state.

30. Exercise!!!

31. 3-SAT is NP-complete