The.Algorithm.Design.Manual.Springer-Verlag.1998

Knapsack Problem
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Knapsack Problem
Input description: A set of items , where item i has size and value . A knapsack
capacity C.
Problem description: Find the subset that maximizes the value of given that ;
i.e. all the items fit in a knapsack of size C.
Discussion: The knapsack problem arises whenever there is resource allocation with financial
constraints. Given a fixed budget, how do you select what things you should buy. Everything has a cost
and value, so we seek the most value for a given cost. The term knapsack problem invokes the image of
the backbacker who is constrained by a fixed-size knapsack and so must fill it only with the most useful
items.
The typical formulation in practice is the 0/1 knapsack problem, where each item must be put entirely in
the knapsack or not included at all. Objects cannot be broken up arbitrarily, so its not fair taking one can
of coke from a six-pack or opening the can to take just a sip. It is this 0/1 property that makes the
knapsack problem hard, for a simple greedy algorithm finds the optimal selection whenever we are
allowed to subdivide objects arbitrarily. For each item, we could compute its ``price per pound'', and
take as much of the most expensive item until we have it all or the knapsack is full. Repeat with the next
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