CSCI 2670 Introduction to Theory of Computing

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

2. Agenda • Yesterday • The Class NP • Value is exponential in length • Today • More on the class NP • Quiz December 1, 2004

3. The class NP Definition: A verifier for a language A is an algorithm V, where A={w|V accepts <w,c> 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. December 1, 2004

4. 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 December 1, 2004

5. 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 December 1, 2004

6. NP P coNP What we know December 1, 2004

7. Are there any problems here? What we don’t know NP P coNP December 1, 2004

8. 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 December 1, 2004

9. NP-completeness • A problem C is NP-complete if finding a polynomial-time solution for C would imply P=NP December 1, 2004

10. 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 December 1, 2004

11. 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 December 1, 2004

12. 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! December 1, 2004

13. 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 December 1, 2004

14. 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. December 1, 2004

15. f f 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 December 1, 2004