Consider the following version of Knapsack. Given are two weight limits W 1 and W 2 , where W 1 ≤ W 2 . Given are also n items (w 1 , c 1 ), (w 2 , c 2 ), . . . , (w n , c n ), where w i is the weight and c i the cost of the i-th item. We want to find a subset of these items of the highest cost, where the subset weights at least W 1 and at most W 2 . Give an O(nW 2 ) algorithm for this problem. (Recall that the cost (respectively weight) of a subset is the sum of the costs (respectively weights) of the items in the subset

Consider the following version of Knapsack. Given are two weight limits W 1 and W 2 , where W 1 ≤ W 2 . Given are also n items (w 1 , c 1 ), (w 2 , c 2 ), . . . , (w n , c n ), where w i is the weight and c i the cost of the i-th item. We want to find a subset of these items of the highest cost, where the subset weights at least W 1 and at most W 2 . Give an O(nW 2 ) algorithm for this problem. (Recall that the cost (respectively weight) of a subset is the sum of the costs (respectively weights) of the items in the subset

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter19: Probabilistic Dynamic Programming

Section: Chapter Questions

Problem 10RP

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Consider the following version of Knapsack. Given are two weight limits W1 and W2, whereW1 ≤ W2. Given are also n items (w1, c1),(w2, c2), . . . ,(wn, cn), where wiis the weight and cithe cost of the i-th item. We want to find a subset of these items of the highest cost, wherethe subset weights at least W1 and at most W2. Give an O(nW2) algorithm for this problem.(Recall that the cost (respectively weight) of a subset is the sum of the costs (respectivelyweights) of the items in the subset.)
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Assume there are n courses offered by the university, where each course has one or no prerequisites.If course j is the prerequisite for course i, then we are only allowed to take course i after course j. And after taking each course i, we can get a reward r_i. Given the reward and prerequisite for each course, design analgorithm to find the maximum total reward we can get by taking m courses. The time complexity shouldbe O(nm^2). Please only answer if you have the algorithm
My question is regarding the knapsack problem. How do i apply the knapsack problem on a table like the given one. In my script it says: " For that put the numbers Opt[k,V] for k=1,..,5 and V= 1,....,9 in a table.For that Opt[k,V] is a partial solution which one obtains for the first k items at maximum weight V receives. I tried to apply the definition of the script on the table, maybe i did it wrong. How do i do it?
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