Parameterized Inapproximability Hypothesis ( \(\textsf{PIH}\) ) is a central question in the field of parameterized complexity. \(\textsf{PIH}\) asserts that given as input a 2- \(\textsf{CSP}\) on k variables and alphabet size n, it is \(\textsf{W}\) [1]-hard parameterized by k to distinguish if the input is perfectly satisfiable or if every assignment to the input violates 1% of the constraints. An important implication of \(\textsf{PIH}\) is that it yields the tight parameterized inapproximability of the \(k\) - \(\textsf{maxcoverage}\) problem. In the \(k\) - \(\textsf{maxcoverage}\) problem, we are given as input a set system, a threshold \(\tau >0\) , and a parameter k and the goal is to determine if there exist k sets in the input whose union is at least \(\tau \) fraction of the entire universe. \(\textsf{PIH}\) is known to imply that it is \(\textsf{W}\) [1]-hard parameterized by k to distinguish if there are k input sets whose union is at least \(\tau \) fraction of the universe or if the union of every k input sets is not much larger than \(\tau \cdot (1-\frac{1}{e})\) fraction of the universe. In this work we present a gap preserving \(\textsf{FPT}\) reduction (in the reverse direction) from the \(k\) - \(\textsf{maxcoverage}\) problem to the aforementioned 2- \(\textsf{CSP}\) problem, thus showing that the assertion that approximating the \(k\) - \(\textsf{maxcoverage}\) problem to some constant factor is \(\textsf{W}\) [1]-hard implies \(\textsf{PIH}\) . In addition, we present a gap preserving \(\textsf{FPT}\) reduction from the \(k\) - \(\textsf{median}\) problem (in general metrics) to the \(k\) - \(\textsf{maxcoverage}\) problem, further highlighting the power of gap preserving \(\textsf{FPT}\) reductions over classical gap preserving polynomial time reductions.
