Assume you are given four objects, and you should fit as many of them into a knapsack of size 15 as possible. The resulting value of the objects should be as large as possible. 1. Please select your objects using the Greedy principle according to some greedy criterion you may choose yourself (please make clear which criterion you’ve used). Is this solution optimal? 2. Please explain and apply the principle of dynamic programming using the example above.

Assume you are given four objects, and you should fit as many of them into a knapsack of size 15 as possible. The resulting value of the objects should be as large as possible. 1. Please select your objects using the Greedy principle according to some greedy criterion you may choose yourself (please make clear which criterion you’ve used). Is this solution optimal? 2. Please explain and apply the principle of dynamic programming using the example above.

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter8: Network Models

Section8.3: Maximum-flow Problems

Problem 8P

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Assume you are given four objects, and you should fit as many of them into a knapsack of size 15as possible. The resulting value of the objects should be as large as possible.1. Please select your objects using the Greedy principle according to some greedy criterion youmay choose yourself (please make clear which criterion you’ve used). Is this solution optimal?2. Please explain and apply the principle of dynamic programming using the example above.
We are given four objects, and we should fit as many of them into a knapsack of size 15as possible. The resulting value of the objects should be as large as possible.1. Select objects using the Greedy principle according to some greedy criterion youmay choose yourself (please make clear which criterion you’ve used). Is this solution optimal?2. Explain and apply the principle of dynamic programming using the example above.
Consider the problem of providing change for an amount of n-bahts while using the smallest possible quantity of coins. You may assume that each coin’s value is integer. Assume that the coins available for use have values that can be expressed as powers of c. To clarify, the denominations of the coins can be written as c0,c1,..., ck , where c is an integer greater than 1, and k is a positive integer. Prove that the greedy algorithm always yields an optimal solution.
Explain how to solve the {0, 1}-knapsack problem using dynamic programming. You are given n objects which cannot be broken into smaller pieces. Moreover, you have only one copy of each object. Each object i (where 1 ≤ i ≤ n) has an integer weight wi > 0 and a value vi > 0. You have a knapsack that can carry a total weight not exceeding W. Your goal is to fill the knapsack in a way that maximizes the total value of the included objects, while respecting the capacity constraint. For each object i (where 1 ≤ i ≤ n), either you bring it or not. 1. Write a recursion for the optimal solution and explain why it is correct. Make sure you define the notation you are using. 2. Consider the following input and fill the table corresponding to the recursion you found in #1: n = 6, w1 = 2, w2 = 2, w3 = 3, w4 = 2, w5 = 5, w6 = 4, v1 = 17, v2 = 2, v3 = 1, v4 = 1, v5 = 18, v6 = 11 and W = 12. Moreover, give all optimal solutions.
If an optimal solution to a problem can be obtained by greedy, It can also be obtained by dynamic programming. True or False?
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a. Given n items, where each item has a weight and a value, and a knapsack that can carry at most W You are expected to fill in the knapsack with a subset of items in order to maximize the total value without exceeding the weight limit. For instance, if n = 6 and items = {(A, 10, 40), (B, 50, 30), (C, 40, 80), (D, 20, 60), (E, 40, 10), (F, 10, 60)} where each entry is represented as (itemIdi, weighti, valuei). Use greedy algorithm to solve the fractional knapsack problem. b. Given an array of n numbers, write a java or python program to find the k largest numbers using a comparison-based algorithm. We are not interested in the relative order of the k numbers and assuming that (i) k is a small constant (e.g., k = 5) independent of n, and (ii) k is a constant fraction of n (e.g., k = n/4). Provide the Big-Oh characterization of your algorithm.
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What is the solution by greedy algorithm to the change-making problem with denominations 1, 3, and 4 to return the minimum number of coins for an amount of 6? Copy and paste the following expression and replace the ? mark by a number ?(4), ?(3), ?(1) (Keep the same format) Is the solution provided by greedy algorithm to the change-making problem with denominations 1, 3, and 4 to return the minimum number of coins for an amount of 6 optimal or not? prove your answer
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