Polynomial formulations as a barrier for reduction-based hardness proofs

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.