Constant Approximating Parameterized $k$-SetCover is W[2]-hard

In this paper, we prove that it is W[2]-hard to approximate $k$-SetCover within any constant ratio. Our proof is built upon the recently developed threshold graph composition technique. We propose a strong notion of threshold graph and use a new composition method to prove this result. Our technique could also be applied to rule out polynomial time $o\left(\frac{\log n}{\log \log n}\right)$ ratio approximation algorithms for the non-parameterized $k$-SetCover problem, assuming W[1]$\ne$FPT. We highlight that our proof does not depend on the well-known PCP theorem, and only involves simple combinatorial objects. Furthermore, our reduction results in a $k$-SetCover instance with $k$ as small as $O\left(\log^2 n\cdot \log \log n\right)$.