The Strong Exponential Time Hypothesis (SETH) asserts that SAT cannot be solved in time $(2-\varepsilon)^n$, where $n$ is the number of variables of an input formula and $\varepsilon>0$ is a constant. Much as SAT is the most popular problem for proving NP-hardness, SETH is the most popular hypothesis for proving fine-grained hardness. For example, the following upper bounds are known to be tight under SETH: $2^n$ for the hitting set problem, $n^k$ for $k$-dominating set problem, $n^2$ for the edit distance problem. It has been repeatedly asked in the literature during the last decade whether similar SETH-hardness results can be proved for other important problems like Chromatic Number, Hamiltonian Cycle, Independent Set, $k$-SAT, MAX-$k$-SAT, and Set Cover. In this paper, we provide evidence that such fine-grained reductions will be difficult to construct. Namely, proving $c^n$ lower bound for any constant $c>1$ (not just $c=2$) under SETH for any of the problems above would imply new circuit lower bounds: superlinear Boolean circuit lower bounds or polynomial arithmetic circuit lower bounds. In particular, it would be difficult to derive Set Cover Conjecture from SETH.

翻译：强烈的光学时间假说(SETH)声称,沙特德士古公司无法及时用美元(2美元)解决沙特德士古公司,美元是投入公式的变量数,美元是不变的。由于沙特德士古公司是证明NP-硬度最流行的问题,赛斯德公司是证明细度硬度的最流行假设。例如,在塞斯克公司之下,已知以下的上界线很紧:撞击设定问题2美元2美元,美元为基值定问题2美元,美元为编辑距离问题2美元。在过去十年里,文献中反复询问,对于其他重要问题,如Chromaticnumber、汉密尔顿循环、独立赛特、美元-SAT、MAX-k美元SAT和Set Cover。在本文中,我们提供证据表明,这种精细度的削减将难以构建。 也就是说,对于任何固定的S-rickral2美元, 将证明S-rickral2美元为低值。