2. One of the biggest open-questions in Computer Science is whether P = NP.

a) What would need to be done to show that P = NP?

b) What would need to be done to show that P NP?

Definition of polynomial-time reducibility

Decision problem A reduces to decision problem B if any instance of problem A, say I_Ai, can be transformed into an instance of problem B, say I_Bj, such that

a) transformation time is a polynomial in the size of I_Ai (call it n_A),

b) the size of I_Bj polynomial w.r.t. to size of I_Ai , and

c) Algorithm B with I_Bj as input answers "yes" if and only if algorithm A with I_Ai as input

answers "yes".

We say "A reduces to problem B," or "A B"

3. Assume there exists a polynomial-time transformation from decision problem A to decision problem B.

a) If we can solve B in poly. time, then how fast can we solve A?

b) If A is known to be "hard", say best worst-case algorithm of (2ⁿ), then what can we conclude about B?