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Partition Problem

Given a finite set A and a ``size'' $s(a) \in Z^+$ for each $a \in A$ , find a subset $A' \subseteq A$ such that

$\begin{displaymath}\sum_{a \in A'} s(a) \; = \; \sum_{a \in (A - A')} s(a)\end{displaymath}$

PARTITION = $\{\langle A, s(a) \rangle: \exists A' \subseteq A$ such that the sums of A' and (A - A') are equal}

For example, if $A = \{a=1, b=2, c=3, d=4, e=5, f=7, g=8\}$ , then one possible partition is $A' = \{a, b, c, d, e\}$ and $A - A' = \{f, g\}$ . The sum of both subsets is 15.

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