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Exercise 1
The two problems are equivalent under the following interpretation:
• The equal due date acts as the capacity of the knapsack;
• The processing times Si acts like the sizes of knapsack items;
• The values Vi of the jobs represents the values (profits) of the knapsack items.
The reduction built this way ensures that selection of the disjoint jobs which lead to maximum profit corresponds to the items which must be put into the knapsack such that to gain maximized profit under the capacity (or deadline, respectively) constraint. Knowing the deadline, Si and Vi we can built a solution to the Knapsack problem. Vice versa, knowing the objects which must be taken into the Knapsack, we use them to build the array of jobs which ensure “the best” achievable profit within deadline....

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