Question: THE QUESTION: Prove that the smart-greedy algorithm 1/2-approximates KNAPSACK. (Hint: What would the optimal value be if we were allowed to take a fraction of

THE QUESTION: Prove that the smart-greedy algorithm 1/2-approximates KNAPSACK. (Hint: What would the optimal value be if we were allowed to take a fraction of each item?)

Background information:

Relatively-greedy algorithm for knapsack: As long as possible to add an item to the KNAPSACK, take the item with the largest relative value: value/weight.

Dumb-greedy algorithm for knapsack: Take the single item with the largest value.

Smart-greedy algorithm for knapsack: Run relatively-greedy and dumb-greedy and pick the best solution among the two.

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