Circuits and Backdoors: Five Shades of the SETH

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.