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Knapsack Problem

KNAPSACK: Given a finite set U, a ``size'' $s(u) \in Z^+$ and a ``value'' $v(u) \in Z^+$ for each $u \in U$ , a size constraint $B \in Z^+$ , and a value goal $K \in Z^+$ , is there a subset $U' \subseteq U$ such that $\sum_{u \in U'} s(u) \leq B$ and $\sum_{u \in U'} v(u) \geq K$ ?

This can be seen as a knapsack, which has a size limit for the objects, as in the picture below.

$\psfig{figure=figures/f27-11.ps}$

The goal is to pick a collection of objects that will fit in the knapsack and whose total value is at least K (K is input)

KNAPSACK = $\{(U, s, v, B, K): \exists$ subset U' of U such that the sum of s values is at most B, and the sum of v values is at least K}

Knapsack Problem Variant

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