np complete.ppt
- 1. David Luebke 1 1/4/2023 CS 332: Algorithms NP Completeness Continued
- 2. David Luebke 2 1/4/2023 Homework 5 ● Extension: due midnight Monday 22 April
- 3. David Luebke 3 1/4/2023 Review: Tractibility ● Some problems are undecidable: no computer can solve them ■ E.g., Turing’s “Halting Problem” ■ We don’t care about such problems here; take a theory class ● Other problems are decidable, but intractable: as they grow large, we are unable to solve them in reasonable time ■ What constitutes “reasonable time”?
- 4. David Luebke 4 1/4/2023 Review: P ● Some problems are provably decidable in polynomial time on an ordinary computer ■ We say such problems belong to the set P ■ Technically, a computer with unlimited memory ■ How do we typically prove a problem P?
- 5. David Luebke 5 1/4/2023 Review: NP ● Some problems are provably decidable in polynomial time on a nondeterministic computer ■ We say such problems belong to the set NP ■ Can think of a nondeterministic computer as a parallel machine that can freely spawn an infinite number of processes ■ How do we typically prove a problem NP? ● Is P NP? Why or why not?
- 6. David Luebke 6 1/4/2023 Review: P And NP Summary ● P = set of problems that can be solved in polynomial time ● NP = set of problems for which a solution can be verified in polynomial time ● P NP ● The big question: Does P = NP?
- 7. David Luebke 7 1/4/2023 Review: NP-Complete Problems ● The NP-Complete problems are an interesting class of problems whose status is unknown ■ No polynomial-time algorithm has been discovered for an NP-Complete problem ■ No suprapolynomial lower bound has been proved for any NP-Complete problem, either ● Intuitively and informally, what does it mean for a problem to be NP-Complete?
- 8. David Luebke 8 1/4/2023 Review: Reduction ● A problem P can be reduced to another problem Q if any instance of P can be rephrased to an instance of Q, the solution to which provides a solution to the instance of P ■ This rephrasing is called a transformation ● Intuitively: If P reduces in polynomial time to Q, P is “no harder to solve” than Q
- 9. David Luebke 9 1/4/2023 An Aside: Terminology ● What is the difference between a problem and an instance of that problem? ● To formalize things, we will express instances of problems as strings ■ How can we express a instance of the hamiltonian cycle problem as a string? ● To simplify things, we will worry only about decision problems with a yes/no answer ■ Many problems are optimization problems, but we can often re-cast those as decision problems
- 10. David Luebke 10 1/4/2023 NP-Hard and NP-Complete ● If P is polynomial-time reducible to Q, we denote this P p Q ● Definition of NP-Hard and NP-Complete: ■ If all problems R NP are reducible to P, then P is NP-Hard ■ We say P is NP-Complete if P is NP-Hard and P NP ■ Note: I got this slightly wrong Friday ● If P p Q and P is NP-Complete, Q is also NP- Complete
- 11. David Luebke 11 1/4/2023 Why Prove NP-Completeness? ● Though nobody has proven that P != NP, if you prove a problem NP-Complete, most people accept that it is probably intractable ● Therefore it can be important to prove that a problem is NP-Complete ■ Don’t need to come up with an efficient algorithm ■ Can instead work on approximation algorithms
- 12. David Luebke 12 1/4/2023 Proving NP-Completeness ● What steps do we have to take to prove a problem P is NP-Complete? ■ Pick a known NP-Complete problem Q ■ Reduce Q to P ○ Describe a transformation that maps instances of Q to instances of P, s.t. “yes” for P = “yes” for Q ○ Prove the transformation works ○ Prove it runs in polynomial time ■ Oh yeah, prove P NP (What if you can’t?)