Complexity
The set packing problem is not only NP-complete, but its optimization version (general maximum set packing problem) has been proven as difficult to approximate as the maximum clique problem; in particular, it cannot be approximated within any constant factor. The best known algorithm approximates it within a factor of . The weighted variant can also be approximated this well.
However, the problem does have a variant which is more tractable: if we assume no subset exceeds k≥3 elements, the answer can be approximated within a factor of k/2 + ε for any ε > 0; in particular, the problem with 3-element sets can be approximated within about 50%. In another more tractable variant, if no element occurs in more than k of the subsets, the answer can be approximated within a factor of k. This is also true for the weighted version.
Read more about this topic: Set Packing
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