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

TL;DR

This paper proves that approximating the parameterized k-SetCover problem within any constant ratio is W[2]-hard, using a novel threshold graph composition technique that avoids reliance on the PCP theorem.

Contribution

Introduces a new threshold graph composition method to establish W[2]-hardness of constant ratio approximation for parameterized k-SetCover, independent of PCP theorem.

Findings

W[2]-hardness of constant ratio approximation for parameterized k-SetCover

New threshold graph composition technique developed

Implications for non-parameterized k-SetCover approximation limits

Abstract

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 graphs 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 with $k$ as small as $O\left(\frac{\log n}{\log \log n}\right)^3$, assuming W[1]$\neq$FPT. We highlight that our proof does not depend on the well-known PCP theorem, and only involves simple combinatorial objects.