Polynomial Time Reductions

1. Polynomial Time Reductions Polynomial Computable function : For any computes in polynomial time

2. Language A is polynomial time reducible to language B if there is a polynomial computable function such that:

3. Theorem: Suppose that is polynomial reducible to . If then . Proof: Let be the machine to accept Machine to accept in polynomial time: On input : 1. Compute 2. Run on input

4. 3CNF formula: Each clause has three literals Language: 3SAT ={ : is a satisfiable 3CNF formula}

5. Clique: A 5-clique CLIQUE = { : contains a -clique}

6. Theorem: 3SAT is polynomial time reducible to CLIQUE Proof: give a polynomial time reduction of one problem to the other

9. NP-Completeness A problem is NP-complete if: • It is in NP • Every NP problem is reduced to it (in polynomial time)

10. Observation: If we can solve any NP-complete problem in Deterministic Polynomial Time (P time) then we know:

11. Observation: If we prove that we cannot solve an NP-complete problem in Deterministic Polynomial Time (P time) then we know:

12. Cook’s Theorem: The satisfiability problem is NP-complete Proof: Convert a Non-Deterministic Turing Machine to a Boolean expression in conjunctive normal form

13. Other NP-Complete Problems: • The Traveling Salesperson Problem • Vertex cover • Hamiltonian Path All the above are reduced to the satisfiability problem