Parameterized Inapproximability Hypothesis (PIH) is a central question in the field of parameterized complexity. PIH asserts that given as input a 2-CSP on $k$ variables and alphabet size $n$, it is 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 PIH is that it yields the tight parameterized inapproximability of the $k$-maxcoverage problem. In the $k$-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. PIH is known to imply that it is 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 FPT reduction (in the reverse direction) from the $k$-maxcoverage problem to the aforementioned 2-CSP problem, thus showing that the assertion that approximating the $k$-maxcoverage problem to some constant factor is W[1]-hard implies PIH. In addition, we present a gap preserving FPT reduction from the $k$-median problem (in general metrics) to the $k$-maxcoverage problem, further highlighting the power of gap preserving FPT reductions over classical gap preserving polynomial time reductions.

翻译：参数化不可近似性假设（PIH）是参数化复杂度领域的核心问题。PIH断言：给定一个包含$k$个变量、字母表大小为$n$的2-CSP实例，以$k$为参数时，判定该实例是完全可满足的还是所有赋值都会违反1%约束的问题是W[1]-难的。PIH的一个重要推论是：它能推导出$k$-最大覆盖问题紧致的参数化不可近似性结果。在$k$-最大覆盖问题中，给定集合系统、阈值$\tau>0$和参数$k$，目标是判断是否存在$k$个输入集合，其并集至少占全集规模的$\tau$比例。已知PIH可推出：以$k$为参数时，区分"存在$k$个输入集合其并集至少占全集$\tau$比例"与"任意$k$个输入集合的并集不超过全集$\tau\cdot (1-\frac{1}{e})$比例"是W[1]-难的。本研究通过构造从$k$-最大覆盖问题到前述2-CSP问题的间隙保持FPT归约（反向），证明"以常数因子近似$k$-最大覆盖问题是W[1]-难的"这一论断可推导出PIH。此外，我们提出了从（一般度量下的）$k$-中值问题到$k$-最大覆盖问题的间隙保持FPT归约，进一步凸显了间隙保持FPT归约相较于经典间隙保持多项式时间归约的优越性。