Circuits and Backdoors: Five Shades of the SETH

The Strong Exponential Time Hypothesis (SETH) is a standard assumption in (fine-grained) parameterized complexity and many tight lower bounds are based on it. We consider a number of reasonable weakenings of the SETH, with sources from (i) circuit complexity (ii) backdoors for SAT-solving (iii) graph width parameters and (iv) weighted satisfiability problems. Our goal is to arrive at formulations which are simultaneously more plausible as hypotheses, but also capture interesting and robust notions of complexity. Using several tools from classical complexity theory we are able to consolidate these numerous hypotheses into a hierarchy of five main equivalence classes of increasing solidity. This framework serves as a step towards structurally classifying a variety of SETH-based lower bounds into intermediate equivalence classes.