Constant Approximating Parameterized k-SetCover is W[2]-hard
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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(log n/loglog n) ratio approximation algorithms for the non-parameterized k-SetCover problem, assuming W[1]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(log^2 n·loglog n).
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