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Set splitting problem
In computational complexity theory, the Set Splitting problem is the following decision problem: given a family ''F'' of subsets of a finite set ''S'', decide whether there exists a partition of ''S'' into two subsets ''S1'', ''S2'' such that all elements of ''F'' are split by this partition, i.e., none of the elements of ''F'' is completely in ''S1'' or ''S2''. Set Splitting is one of Garey&Johnson's classical NP-complete problems.〔
== Variants ==
The optimization version of this problem is called Max Set Splitting and requires finding the partition which maximizes the number of split elements of ''F''. It is an APX-complete〔 (and hence NPO-hard) problem.
When each element of ''F'' is restricted to be of cardinality exactly ''k'', the decision variant is called E''k''-Set Splitting and the optimization version Max E''k''-Set Splitting. For ''k'' ≥ 2, the former remains NP complete and the latter APX complete.〔
The Weighted Set Splitting is a variant in which the subsets in ''F'' have weights and the objective is to maximize the total weight of the split subsets.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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