Lecture 6 - What is NP-Complete? Formally?

Question 1

When is a decision problem NP complete?

Answer 1

• if the problem is in the NP class
• every problem can be ptred to the problem

Question 2

Consequences of the definition?

Answer 2

• if problem was to be NP-complete and in P then all NP problems would be in NP
• every problem in NP can be solved in polynomial time by PTR to another problem
• if P is not equal to NP and problem is NP-complete, then problem is not in P

Question 3

How do we prove a problem is NP-Complete?

Answer 3

• show it is in NP
• there exists a PTR to it from an NP problem

Question 4

What is a restriction of a problem?

Answer 4

A subset of instances of that problem.

Question 5

If the restriction of the problem is Np-Complete is the whole problem Np-Complete?

Answer 5

YES

Question 6

Given an NP-Complete problem, is the restriction NP-COMPLETE?

Answer 6

Not necessairly

Question 7

Is 2-GCP Np-Complete?

Answer 7

No, it is in P

Question 8

Is 3-GCP NP-COMPLETE?

Answer 8

YES

Question 9

Is 2-SAT NP-COMPLETE?

Answer 9

No, it is in P

Question 10

Is 3-SAT NP-COMPLETE?

Answer 10

YES

• can be proven by SAT -> 3 SAT via PTR

Question 11

What to do when faced with an Np-Complete problem?

Answer 11

• maybe only focus on a restricted version
• approximation alg (for optimisation)
• probabilistic alg (for decision)
• exponential time alg rather than exhaustive search