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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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