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CS 332: Algorithms

CS 332: Algorithms. NP Completeness Continued. Homework 5. Extension: due midnight Monday 22 April. 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

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CS 332: Algorithms

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  1. CS 332: Algorithms NP Completeness Continued David Luebke 13/11/2014

  2. Homework 5 • Extension: due midnight Monday 22 April David Luebke 23/11/2014

  3. 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”? David Luebke 33/11/2014

  4. 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? David Luebke 43/11/2014

  5. 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? David Luebke 53/11/2014

  6. 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? David Luebke 63/11/2014

  7. 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? David Luebke 73/11/2014

  8. 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 David Luebke 83/11/2014

  9. 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 David Luebke 93/11/2014

  10. 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 alsoNP- Complete David Luebke 103/11/2014

  11. 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 David Luebke 113/11/2014

  12. Proving NP-Completeness • What steps do we have to take to prove a problem Pis 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?) David Luebke 123/11/2014

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