Final answer:
The best technique to solve the problem of finding a subset of items for maximum profit within a weight limit - the Knapsack Problem - is Dynamic Programming. It breaks down the problem, stores results of subproblems, and builds towards the optimal solution in a comprehensive manner. Option 2 is the correct answer.
Step-by-step explanation:
The student's question involves finding a subset of items that yields maximum profit without exceeding a given total weight limit. This is a classic problem known as the Knapsack Problem, which is a fundamental issue in combinatorial optimization. The best solution technique for this problem is to use Dynamic Programming, an approach that solves problems by breaking them down into simpler subproblems and storing the results of these subproblems to avoid redundant work.
While a Greedy Algorithm may appear to be a straightforward approach, it doesn't always lead to the optimal solution for the Knapsack Problem because it makes local optimizations that may miss the overall best combination of items. Backtracking is a more exhaustive technique which explores various combinations but can be less efficient, and Brute Force involves checking every possible combination, which becomes infeasible for larger numbers of items due to its exponential time complexity.
Applying Dynamic Programming involves creating a table where the rows represent items and columns represent weight capacities up to the given limit 'C'. This table is then filled in iteratively, considering whether to include each item in the optimal set for each weight capacity based on maximizing profit without exceeding the weight. This way, the technique ensures that at each step, we are working towards the optimal solution, culminating in the overall maximum profit for the weight constraint 'C'.
The final answer for the best technique to solve this problem is Dynamic Programming.