  The Strong Exponential Time Hypothesis (SETH) asserts that for every
$\varepsilon>0$ there exists $k$ such that $k$-SAT requires time
$(2-\varepsilon)^n$. The field of fine-grained complexity has leveraged SETH to
prove quite tight conditional lower bounds for dozens of problems in various
domains and complexity classes, including Edit Distance, Graph Diameter,
Hitting Set, Independent Set, and Orthogonal Vectors. Yet, it has been
repeatedly asked in the literature whether SETH-hardness results can be proven
for other fundamental problems such as Hamiltonian Path, Independent Set,
Chromatic Number, MAX-$k$-SAT, and Set Cover.
  In this paper, we show that fine-grained reductions implying even
$\lambda^n$-hardness of these problems from SETH for any $\lambda>1$, would
imply new circuit lower bounds: super-linear lower bounds for Boolean
series-parallel circuits or polynomial lower bounds for arithmetic circuits
(each of which is a four-decade open question).
  We also extend this barrier result to the class of parameterized problems.
Namely, for every $\lambda>1$ we conditionally rule out fine-grained reductions
implying SETH-based lower bounds of $\lambda^k$ for a number of problems
parameterized by the solution size $k$.
  Our main technical tool is a new concept called polynomial formulations. In
particular, we show that many problems can be represented by relatively
succinct low-degree polynomials, and that any problem with such a
representation cannot be proven SETH-hard (without proving new circuit lower
bounds).
