  The SETH is a hypothesis of fundamental importance to (fine-grained)
parameterized complexity theory and many important tight lower bounds are based
on it. This situation is somewhat problematic, because the validity of the SETH
is not universally believed and because in some senses the SETH seems to be
"too strong" a hypothesis for the considered lower bounds. Motivated by this,
we consider a number of reasonable weakenings of the SETH that render it more
plausible, with sources ranging from circuit complexity, to backdoors for
SAT-solving, to graph width parameters, to weighted satisfiability problems.
Despite the diversity of the different formulations, we are able to uncover
several non-obvious connections using tools from classical complexity theory.
This leads us to a hierarchy of five main equivalence classes of hypotheses,
with some of the highlights being the following:
  We show that beating brute force search for SAT parameterized by a modulator
to a graph of bounded pathwidth, or bounded treewidth, or logarithmic
tree-depth, is actually the same question, and is in fact equivalent to beating
brute force for circuits of depth $\epsilon n$; we show that beating brute
force search for a strong 2-SAT backdoor is equivalent to beating brute force
search for a modulator to logarithmic pathwidth; we show that beting brute
force search for a strong Horn backdoor is equivalent to beating brute force
search for arbitrary circuit SAT.
