We show, assuming the (randomized) Gap Exponential Time Hypothesis (Gap-ETH), that the following tasks cannot be done in T(k) center dot N-o(k)-time for any function T where N denote the input size: (1 - 1/e + epsilon)-approximation for Max k-Coverage for any constant epsilon > 0, (1 + 2/e-epsilon)-approximation for k-Median (in general metrics) for any constant epsilon > 0. (1 + 8/e-epsilon)-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 all problems considered here can be trivially solved in N-O(k) time, our running time lower bounds are tight up to a constant factor in the exponent. In terms of approximation ratios, Max k-Coverage is well-known to admit polynomial-time (1 - 1/e)-approximation algorithms, and, recently, it was shown that k-Median and k-Mean are approximable to within factors of (1 + 2/e) and (1 + 8/e) respectively in FPT time [20]; hence, our inapproximability ratios are also tight for these three problems. For the remaining problems, no non-trivial FPT approximation algorithms are known. respectively. With this hardness, the above results follow immediately from known reductions. The hardness of Label Cover is in turn shown via a t-wise agreement testing theorem of the following form: given local boolean functions f(1) ,..., fk on domains S-1,...,S-k subset of [n], if random t functions "weakly agree" with sufficiently large probability, then we can find a global boolean function g : [n] -> {0, 1} that "mostly agrees" with "many" of the local functions. We prove such a statement in the regime where S-1,...,Sk are "random-looking" sets of size Theta(n/k).The starting point of all our hardness results is the Label Cover problem (with projection constraints). We show that Label Cover cannot be approximated to within any constant factor in T(k) . 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
