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I know that the Set-cover problem with $n$ elements and a universe of size $N$ is NP complete. Also, the problem is has parameterized complexity regarding the number of sets $k$ that should cover the whole universe: There are only ${n \choose k}=\mathcal{O}(n^k)$ different combinations of $k$ subsets.

My specific problem is easily reducible to the set-cover problem with $k=3$. My approach that iterates over all ${n \choose k}=\mathcal{O}(n^3)$ subsets does not perform well enough.

Is there any specialized algorithm for the 3-set-cover problem? Ideally, I would like some parameterized algorithm or some potent preprocessing techniques that exploit $k=3$.

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    $\begingroup$ You might be able to go down to $O(n^2)$ using a meet-in-the-middle approach, but it seems hard to go beyond that. $\endgroup$ – Yuval Filmus Oct 19 '18 at 15:52
  • $\begingroup$ An algorithm running in time $n^k$ is not FPT for the parameter $k$, it is only an XP-algorithm. You need a running time of the form $f(k) n^{O(1)}$. $\endgroup$ – Juho Oct 19 '18 at 17:34
  • $\begingroup$ @juho true, I remembered incorrectly $\endgroup$ – Samuel Pilz Oct 20 '18 at 7:04
  • $\begingroup$ @YuvalFilmus I would be very interested in a $\mathcal{O}(n^2)$ algorithm. I will try some meet-in-the-middle approach. Do you have a $\mathcal{O}(n^2)$ solution? $\endgroup$ – Samuel Pilz Oct 20 '18 at 7:09
  • $\begingroup$ I don’t have such a solution in mind. $\endgroup$ – Yuval Filmus Oct 20 '18 at 8:57

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