We consider questions that arise from the intersection between the areas of polynomial-time approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms. The questions, which have been asked several times (e.g., [Marx08, FGMS12, DF13]), are whether there is a non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and Minimum Dominating Set (DomSet) problems parameterized by the size of the optimal solution. In particular, letting $text{OPT}$ be the optimum and $N$ be the size of the input, is there an algorithm that runs in $t(text{OPT})text{poly}(N)$ time and outputs a solution of size $f(text{OPT})$, for any functions $t$ and $f$ that are independent of $N$ (for Clique, we want $f(text{OPT})=omega(1)$)? In this paper, we show that both Clique and DomSet admit no non-trivial FPT-approximation algorithm, i.e., there is no $o(text{OPT})$-FPT-approximation algorithm for Clique and no $f(text{OPT})$-FPT-approximation algorithm for DomSet, for any function $f$ (e.g., this holds even if $f$ is the Ackermann function). In fact, our results imply something even stronger: The best way to solve Clique and DomSet, even approximately, is to essentially enumerate all possibilities. Our results hold under the Gap Exponential Time Hypothesis (Gap-ETH) [Dinur16, MR16], which states that no $2^{o(n)}$-time algorithm can distinguish between a satisfiable 3SAT formula and one which is not even $(1 - epsilon)$-satisfiable for some constant $epsilon > 0$. Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for Maximum Balanced Biclique, Maximum Subgraphs with Hereditary Properties, and Maximum Induced Matching in bipartite graphs. Additionally, we rule out $k^{o(1)}$-FPT-approximation algorithm for Densest $k$-Subgraph although this ratio does not yet match the trivial $O(k)$-approximation algorithm.
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