CS 5315, Test 2, Monday, April 30, 2012


(Please do not forget to put your name on all extra sheets of paper)

1. Define what is P, what is NP, what is NP-hard, and what is NP-complete; no need to give a detailed definition of reduction. Which of these three classes contain the problem of adding all numbers in a given array? the 3-coloring problem?

2-6. Reduce the satisfiability problem for the formula (~x1 \/ x2) & (x1 \/ ~x2) & (~x1 \/ x2 \/ x3) to:

Use a greedy algorithm to find a solution to the original satisfiability problem.

7-9. A straightforward algorithm takes time O(n) to compute the sum of all the elements in an array of size n.

10. If a probabilistic algorithm produces a result with the probability of error 1/2, how many times do we need to repeat it to reduce the probability of error to 1%?

11. Find what is Π4PΣ2P.

12. Which class of the polynomial hierarchy corresponds to winning in two steps -- i.e., after you make your move, whatever move the opponent makes, you win.