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Recent years have seen great progress in the approximability of fundamental clustering and facility location problems on high-dimensional Euclidean spaces, including k-Means and k-Median. While they admit strictly better approximation ratios than their general metric versions, their approximation ratios are still higher than the hardness ratios for general metrics, leaving the possibility that the ultimate optimal approximation ratios will be the same between Euclidean and general metrics. Moreover, such an improved algorithm for Euclidean spaces is not known for Uncapaciated Facility Location (UFL), another fundamental problem in the area. In this paper, we prove that for any γ ≥ 1.6774 there exists ε > 0 such that Euclidean UFL admits a (γ, 1 + 2e^{-γ} - ε)-bifactor approximation algorithm, improving the result of Byrka and Aardal [Byrka and Aardal, 2010]. Together with the (γ, 1 + 2e^{-γ}) NP-hardness in general metrics, it shows the first separation between general and Euclidean metrics for the aforementioned basic problems. We also present an (α_Li - ε)-(unifactor) approximation algorithm for UFL for some ε > 0 in Euclidean spaces, where α_Li ≈ 1.488 is the best-known approximation ratio for UFL by Li [Li, 2013]. 

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 τ > 0, and a parameter k and the goal is to determine if there exist k sets in the input whose union is at least τ 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 τ fraction of the universe or if the union of every k input sets is not much larger than τ⋅ (1-1/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.