Tight Running Time Lower Bounds for Strong Inapproximability of Maximum $k$-Coverage, Unique Set Cover and Related Problems (via $t$-Wise Agreement Testing Theorem)

We show, assuming the (randomized) Gap Exponential Time Hypothesis (Gap-ETH), that the following tasks cannot be done in $T(k) \cdot N^{o(k)}$-time for any function $T$ where $N$ denote the input size: - $\left(1 - \frac{1}{e} + \epsilon\right)$-approximation for Max $k$-Coverage for any $\epsilon > 0$, - $\left(1 + \frac{2}{e} - \epsilon\right)$-approximation for $k$-Median (in general metrics) for any constant $\epsilon > 0$. - $\left(1 + \frac{8}{e} - \epsilon\right)$-approximation for $k$-Mean (in general metrics) for any constant $\epsilon > 0$. - Any constant factor approximation for $k$-Unique Set Cover, $k$-Nearest Codeword Problem and $k$-Closest Vector Problem. - $(1 + \delta)$-approximation for $k$-Minimum Distance Problem and $k$-Shortest Vector Problem for some $\delta > 0$. Since these problems can be trivially solved in $N^{O(k)}$ time, our running time lower bounds are essentially tight. In terms of approximation ratios, Max $k$-Coverage is well-known to admit polynomial-time $\left(1 - \frac{1}{e}\right)$-approximation algorithms, and, recently, it was shown that $k$-Median and $k$-Mean are approximable to within factors of $\left(1 + \frac{2}{e}\right)$ and $\left(1 + \frac{8}{e}\right)$ respectively in FPT time [Cohen-Addad et al. 2019]; hence, our inapproximability ratios are also tight for these three problems. For the remaining problems, no non-trivial FPT approximation algorithms are known. The starting point of all our hardness results mentioned above is the Label Cover problem (with projection constraints). We show that Label Cover cannot be approximated to within any constant factor in $T(k) \cdot N^{o(k)}$ time, where $N$ and $k$ denote the size of the input and the number of nodes on the side with the larger alphabet respectively. With this hardness, the above results follow immediately from known reductions...