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The set packing problem is : Given a universe $U$ and a family $S$ of subsets of $U$, a packing is a subfamily $C\subseteq S$ of sets such that all sets in $C$ are pairwise disjoint, and the size of the packing is $|C|$. The goal is to find the $C$ where $|C|$ is maximum.

According to Wikipedia, there exists a $\sqrt{|U|}$-approximation algorithm for this problem. But it doesn't give a way to approach that, it seems that the way cannot be found on google as well.

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    $\begingroup$ The notation $\{S\}$ means "The set whose only element is $S$" not "$S$ is a set." $S$ and $\{S\}$ are completely different objects: you can think of $S$ as a box containing things; $\{S\}$ is a box containing a box containing those things. $\endgroup$ – David Richerby Dec 2 '15 at 8:25
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See Viggo Kann's page on set packing. Viggo mentions two $\sqrt{|U|}$ algorithms:

  • Unweighted case: Halldórsson, M. M., Kratochvíl, J., and Telle, J. A. (1998), "Independent sets with domination constraints", Proc. 25th Int. Colloquium on Automata, Languages and Programming, Lecture Notes in Comput. Sci. 1443, Springer-Verlag, 176-185.

  • Weighted case: Halldórsson, M. M. (1999a), "Approximations of weighted independent set and hereditary subset problems", Proc. 5th Ann. Int. Conf. on Computing and Combinatorics, Lecture Notes in Comput. Sci., Springer-Verlag, 261-270.

I found this link on the first page of the Google search set packing approximation algorithm. I suggest that next time you spend some quality time with your favorite search engine, looking at all links in the first few pages.

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