What are the different algorithms we may use to solve the 0/1 Knapsack problem? Compare and...

What are the different algorithms we may use to solve the 0/1 Knapsack problem? Compare and contrast the different methods.

Expert Solution

We can use Brute Force, Dynamic Programming, Memory Functions, Branch and Bound, and Greedy Algorithms to solve the Knapsack Problem where one has to maximize the benefit of items in a knapsack without extending its capacity.

Testing I: Increase the number of items & Capacity = 50

The comparative study of the brute force, greedy, dynamic programming, branch and bound and genetic algorithms shows that while the complexities of these algorithms are known, the nature of the problem they are applied to makes some of them more suitable than others. The best approximation approaches for the 0/1 Knapsack Problem are dynamic programming and genetic algorithms. As we have shown, the choice between the two depends on the capacity of the knapsack and the size of the population. However, one may decide to choose dynamic programming over genetic algorithms in any circumstances, because it is easy and straightforward to code. In contrast, genetic algorithms require a lot more time in terms of understanding the concepts of the paradigm and in terms of programming effort.

venereology answered 1 year ago

Recall the dynamic programming algorithm we saw in class for solving the 0/1 knapsack problem for...

Recall the dynamic programming algorithm we saw in class for solving the 0/1 knapsack problem for n objects with a knapsack capacity of K. In particular, we characterized our recurrence OPT(j, W) to be following quantity: OPT(j, W) := The maximum profit that can be obtained when selecting from objects 1, 2, . . . , j with a knapsack capacity of W , where (after filling in our dynamic programming table), we return the value stored at OPT(n, K)...

In your own words describe the ‘knapsack problem’. Further, compare and contrast the use of a...

In your own words describe the ‘knapsack problem’. Further, compare and contrast the use of a brute force and a dynamic programming algorithm to solve this problem in terms of the advantage and disadvantages of each. An analysis of the asymptotic complexity of each is required as part of this assignment.

Urgent! How do you do solve a knapsack problem recursively in JAVA. for an arbitrary knapsack...

Urgent! How do you do solve a knapsack problem recursively in JAVA. for an arbitrary knapsack capacity and series of weights. Assume the weights are sorted in an array. **The arguments to the knapsack function are target weight and the array index where the remaining items start. For example if you want your knapsack to weigh exactly 20 pounds and you have five items with weights of 11,8,7,6 and 5 pounds. In this case only combination that works is 8,...

In C++ or Java Write the Greedy programming Algorithm for the 0-1 Knapsack problem. (a)...

In C++ or Java Write the Greedy programming Algorithm for the 0-1 Knapsack problem. (a) No fraction allowed Max Weight W= 9 item 1: profit $20, weight 2, prof/weight=$10 item 2: profit $30, weight 5, prof/weight=$6 item 3: profit $35, weight 7, prof/weight=$5 item 4: profit $12, weight 3, prof/weight=$4 item 5: profit $3, weight 1, prof/weight=$3

Provide the dynamic-programming recurrence for a function that is used to solve 0-1 Knapsack. Clearly define...

Provide the dynamic-programming recurrence for a function that is used to solve 0-1 Knapsack. Clearly define what the function is computing.

2. Use the Laplace transform to solve the initial value problem. ?"+4?=?(?), ?(0)=1, ?′(0)=−1 = {...

2. Use the Laplace transform to solve the initial value problem. ?"+4?=?(?), ?(0)=1, ?′(0)=−1 = { 1, ? < 1 where ?(?) = {0, ? > 1.

Prove that the 0/1 KNAPSACK problem is NP-Hard. (One way to prove this is to prove...

Prove that the 0/1 KNAPSACK problem is NP-Hard. (One way to prove this is to prove the decision version of 0/1 KNAPSACK problem is NP-Complete. In this problem, we use PARTITION problem as the source problem.) (a) Give the decision version of the O/1 KNAPSACK problem, and name it as DK. (b) Show that DK is NP-complete (by reducing PARTITION problem to DK). (c) Explain why showing DK, the decision version of the O/1 KNAPSACK problem, is NP-Complete is good...

1. Use the Laplace transform to solve the initial value problem. ?"+4?′+3?=1−?(?−2)−?(?−4)+?(?−6), ?(0)=0, ?′(0)=0 2. Use...

1. Use the Laplace transform to solve the initial value problem. ?"+4?′+3?=1−?(?−2)−?(?−4)+?(?−6), ?(0)=0, ?′(0)=0 2. Use the Laplace transform to solve the initial value problem. ?"+4?=?(?), ?(0)=1, ?′(0)=−1 = { 1, ? < 1 where ?(?) = {0, ? > 1.

Use a Laplace transform to solve the initial value problem: y''' =y+1, y(0) = 0, y'(0)...

Use a Laplace transform to solve the initial value problem: y''' =y+1, y(0) = 0, y'(0) = 0, y''(0) = 0.

discuss the differences and similarities (implementation and performance) between the solution for 0-1 knapsack problem using...

discuss the differences and similarities (implementation and performance) between the solution for 0-1 knapsack problem using Backtracking versus Branch and Bound. Do not just define the algorithms compare the performance and implementation of the 0-1 knapsack problem

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