1. For each of the problems below, if it is solvable in polynomial time in the worst case,
then prove it. You do not have to solve it.
If it is NP-Complete, then prove it. You may assume they are in class NP.
Also you can use any of Bin Packing, Maximum Cliques, Graph Coloring, Knapsack, CNF SAT,
Vertex Cover, Set Cover and 3 Dimensional Matching as a known NP-Complete problem for your proof.
Problem A: There are m groups of people and they are to attend meetings held in n
rooms. No group is allowed to send more than one of its members to any single meeting.
For each i there are p(i) people in group i
and room j can hold q(j) people.
Find an assignment of the people to the rooms. [33 points]
Problem B: There are m groups of people and they are to attend meetings held in n
rooms. No group is allowed to send more than one of its members to any single meeting.
In addition, certain pairs of people can not be assigned to any room together.
For each i there are p(i) (
) people in group i
and all rooms can hold as many people as necessary.
Find an assignment of the people to the rooms if p(i) 
q for each i. [33 points]
2. Formulate the following problem as a LP problem and solve it by the simplex method:
Any food is good for your health if appropriate amount is taken. But any excess is usually harmful.
Consider two types of food for simplicity: type A and type B. Type A produces 3 kcal of energy per serving per day.
Type B produces 4 kcal of energy per serving
if two or less servings are taken per day. If more than two servings are taken, you lose energy
at the rate of
1 kcal per serving.
Due to the balance required between foods, the total servings of type A can not exceed
that of type B plus 1 serving per day. Type A costs $3.00 per serving and type B $2.00.
Find the amount of servings
to maximize the total energy without spending more than $7.00 a day. [34 points]