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 $k=3$.
