1. CSCI 2670 Introduction to Theory of Computing December 1, 2004

2. 2 Agenda Yesterday –The Class NP –Value is exponential in length Today –More on the class NP –Quiz

3. December 1, 20043 Announcement

4. December 1, 20044 The class NP Definition: A verifier for a language A is an algorithm V, where A={w|V accepts for some string c} The string c is called a certificate of membership in A. Definition: NP is the class of languages that have polynomial-time verifiers.

5. December 1, 20045 Is NP closed under complementation? For example, can we verify in polynomial time that a graph cannot be 3-colored? –Not obviously –It seems we need to check many 3- colorings before we can conclude that none exist The 3-coloring problem is in coNP

6. December 1, 20046 Who wants $1,000,000? In May, 2000, the Clay Mathematics Institute named seven open problems in mathematics the Millennium Problems –Anyone who solves any of these problems will receive $1,000,000 –Proving whether or not P equals NP is one of these problems

7. December 1, 20047 What we know NP PcoNP

8. December 1, 20048 What we don’t know NP PcoNP Are there any problems here?

9. December 1, 20049 Solving NP problems The best-known methods for solving problems in NP that are not known to be in P take exponential time –Brute force search We don’t know if NP is actually in a smaller complexity class

10. December 1, 200410 NP-completeness A problem C is NP-complete if finding a polynomial-time solution for C would imply P=NP

11. December 1, 200411 An NP-complete problem A formula is Boolean if each of its variables can be assigned the values TRUE (1) or FALSE (0) A Boolean formula is satisfiable if there is some assignment of values that results in the formula evaluating to TRUE SAT: Is a given Boolean formula satisfiable? SAT is NP-complete

12. December 1, 200412 Examples (x  y)  (  x  y) –Satisfiable – e.g., x = y = 1 ((x  y)  (  x  z))  ((  x  y)  (  y  z)) –Satisfiable – e.g., x = 0, y = z = 1 ((x  y)  (  x  z))  ((x  y)  (  y  z)) –Unsatisfiable

13. December 1, 200413 Proving a problem is NP-complete A problem C is NP-complete if finding a polynomial-time solution for C would imply P=NP If a polynomial-time solution is found for C, then that solution can be used to find a polynomial-time solution for any other problem in NP –What does this remind you of? –Reductions!

14. December 1, 200414 Reductions and NP-completeness If we can prove an NP-complete problem C can be polynomially reduced to a problem A, then we’ve shown A is NP-complete –A polynomial-time solution to A would provide a polynomial-time solution to C, which would imply P=NP

15. December 1, 200415 Polynomial functions Definition: A function f:Σ *  Σ * is a polynomial time computable function if some polynomial time Turing machine M exists that halts with just f(w) on its tape, when started on any input w.

16. December 1, 200416 Polynomial reductions Definition: Language A is polynomial- time reducible to language B, written A ≤ P B, if a polynomial time computable function f:Σ *  Σ * exists, where for every w w  A iff f(w)  B f f