CS 331 Design and Analysis of Algorithms Homework #4 | Complete Solution

CS 331 Design and Analysis of Algorithms Homework #4

(Total: 100 points)
1. (25 points) Given A=[252], B=[220], C=[2010], and D=[105]. Compute the C(i,j)'s defined in the class for all i < j and give the solution (i.e. the multiplication order) for this instance of Chained Matrix Multiplication.

2. (25 points) Given the following input instance for the 0/1 Knapsack problem. Show how the algorithm given in the class works on this input instance. Complete the whole (M+1)(n+1) table, and give the answer for the optimal solution, where M is the capacity of the knapsack and n is the number of objects. Object 1 Object 2 Object 3 Object 4 Object 5 Object 6
Profit pi 3 3 5 5 1 4
Weight wi 2 4 5 3 2 5
M = 10

3. (26 points) Consider the 0/1/2/3 Knapsack Problem. Unlike 0/1 Knapsack problem which restricts xi to be either 0 or 1, 0/1/2/3 Knapsack Problem allows xi to be either 0 or 1 or 2 or 3 (that is, we assume that 3 copies of each object i are available, for all i). (a) Obtain the dynamic programming functional equation to solve the 0/1/2/3 Knapsack
Problem.
(b) Give an algorithm to implement your functional equation.
(c) What is the complexity of your algorithm?

4. (24 points) Decide True or False for each of the followings. You MUST briefly justify your answer.
(a) If P = NP, then all problems in NP can be solved in polynomial time deterministically.
(b) It is known that 3SAT is NP-complete. Assume that an O(n7413) deterministic
algorithm has been found for the 3SAT problem, then P = NP.
(c) If a decision problem A is NP-complete, proving that A is reducible to B, in
polynomial time, is sufficient to show that B is NP-complete.