1. Lecture 2-2 NP Class

2. P = ? NP = ? PSPACE
They are central problems in computational complexity.

3. If P = NP, then
NP-complete P

4. Ladner Theorem
If NP ≠ P, then there exists a set A lying -between P and NP-complete class, i.e., A is in NP, but not in P and not being NP-compete.

5. Is it true that
a problem belongs to NP iff
its solution can be polynomial-time
checkable ?
Answer: No!

6. Integer Programming

7. Decision version of
Integer Programming

9. How to prove a decision problem belonging to NP?
How to design a polynomial-time nondeterministic algorithm?

10. Hamiltonian Cycle Given a graph G, does G contain a Hamiltonian cycle?
Hamiltonian cycle is a cycle passing every vertex exactly once.

11. Post office

12. Nondeterministic Algorithm
Guess a permutation of all vertices.
Check whether this permutation gives a cycle. If yes, then algorithm halts.
What is the running time?

13. Minimum Spanning Tree
Given an edge-weighted graph G, find a spanning tree with minimum total weight.
Decision Version: Given an edge-weighted graph G and a positive integer k, does G contains a spanning tree with total weight < k.

14. Nondeterministic Algorithm
Guess a spanning tree T.
Check whether the total weight of T < k.
This is not clear!

15. How to guess a spanning tree?
Guess n-1 edges where n is the number of vertices of G.
Check whether those n-1 edges form a connected spanning subgraph, i.e., there is a path between every pair of vertices.

16. Co-decision version of MST
Given an edge-weighted graph G and a positive integer k, does G contain no spanning tree with total weight < k?

17. Algorithm Computer a minimum spanning tree.
Check whether its weight > k. If yes, the algorithm halts.

18. co-NP
co-NP = {A | Σ* - A ε NP}

19. NP ∩ co-NP
So far, no natural problem has been found in NP ∩ co-NP, but not in P. NP
co-NP P

20. Linear Programming
Decision version: Given a system of linear inequality, does the system have a solution?
It was first proved in NP ∩ co-NP and later found in P (1979).

21. Primality Test Given a natural number n, is n a prime?
It was first proved in NP ∩ co-NP and later found in P (2004).

22. Therefore
A natural problem belonging to NP ∩ co-NP is a big sign for the problem belonging to P.

23. Proving a problem in NP In many cases, it is not hard.
In a few cases, it is not easy.

24. Integer Programming
Decision version: Given A and b, does Ax > b contains an integer solution?
The difficulty is that the domain of “guess” is too large.

25. Polynomial-time many-one reduction

26. 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.

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

28. 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?

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

30. 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?

31. Boolean Algebra

32. Boolean Algebra

33. 3CNF

34. Examples

36. 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?

37. Examples
Not a polynomial-time
reduction!!!

42. 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

43. 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

44. Characterization of NP

46. Thanks, end