1. a. Apply the bottom-up dynamic programming algorithm to the following instance of the knapsack problem: item 1 23 45 weight 3 2 1 4 5 value $25 $20 $15 $40 $50 capacity W = 6. b. How many different optimal subsets does the instance of part (a) have? c. In general, how can we use the table generated by the dynamic program- ming algorithm to tell whether there is more than one optimal subset for the knapsack problem's instance?

8. A school is creating class schedules for its students. The students submit their requested courses and then a program will be designed to find the optimal schedule for all students. The school has determined that finding the absolute best schedule cannot be solved in a reasonablo time. Instead they have decided to use a simpler algorithm that produces a good but non-optimal schedule in a more reasonable amount of time. Which principle does this decision best demonstrate? O A Unreasonable algorithms may sometimes also be undecidable O B. Heuristics can be used to solve some problems for which no reasonable algorithm exsts O C. efficiency Two algorithms that solve the same problem must also have the same O B. Approximate solutions are often identical to optimal solutions

A student has been asked to put some parcels on a shelf. The parcels all weigh different amounts, and the shelf has a maximum safe loading weight capacity of 150 Kg. The weight of parcels are as follows (in Kg): The student has been asked to load the maximum weight possible parcels on the shelf subject to the maximum safe loading weight. State two possible approaches for a greedy algorithm solution to solve this problem. In each case, state clearly the result you would get from applying that approach to this problem, stating whether the solution is optimal or not. If your answer does not produce an optimal solution, what algorithm could be employed to find one?

The monotone restriction (MR) on the heuristic function is defined as h (nj ) 2 h (ni ) - c (ni , nj ). Please prove the following: 1. If h(n)