The Exponential Time Hypothesis and the Parameterized Clique Problem

In parameterized complexity there are three natural definitions of fixed-parameter tractability called strongly uniform, weakly uniform and nonuniform fpt. Similarly, there are three notions of subexponential time, yielding three flavours of the exponential time hierarchy (ETH) stating that 3Sat is not solvable in subexponential time. It is known that ETH implies that p-Clique is not fixed-parameter tractable if both are taken to be strongly uniform or both are taken to be uniform, and we extend this to the nonuniform case. We also show that even weakly uniform subexponential time is strictly contained in nonuniform subexponential time. Furthermore, we deduce from nonuniform ETH that no single exponent d allows for arbitrarily good fpt-approximations of clique.