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?

3. If we want to show that the clique-decision problem is NP-complete, then we must show that

the clique-decision problem is in NP, and CNF - SAT reduces to the clique-decision problem.

CNF-SAT problem reduces to Clique-decision problem if any Boolean expression B in CNF can be transformed into an instance of clique-decision problem such that

a) transformation time is a polynomial in the size of the Boolean expression

b) the size of clique-decision problem a polynomial w.r.t. to size of the Boolean expression, and

c) clique-decision problem answers "yes" if and only if the CNF Boolean expression is satisfiable.

Transform CNF expression B into a graph G = (V, E) by

V = { (y, i) such that y is a literal in clause c_i }, i.e., a node for each literal within every clause

E = { ((y, i), (z, j)) such that i j and }, i.e., edges between all nodes, except between nodes corresponding to literals

• within the same clause, and
• literals that conflict.

a) Argue that the transformation time is a polynomial in the size of the Boolean expression

b) the size of clique-decision problem a polynomial w.r.t. to size of the Boolean expression, and

c) To show that the clique-decision problem answers "yes" if and only if the CNF Boolean expression is satisfiable. We must show that

i) if B is satisfiable, then G has clique of size k, and

ii) if G has a clique of size k, then B is satisfiable.