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.

翻译：SETH（强指数时间假说）对于（细粒度）参数化复杂度理论具有基础性重要性，许多关键的紧下界都基于该假说。然而这一现状存在一定问题，一方面SETH的有效性并未获得普遍认同，另一方面在某些意义上SETH对于所考虑的下界而言似乎是"过强"的假说。受此驱动，我们考虑了一系列使SETH更可信的合理弱化形式，其来源涵盖电路复杂度、SAT求解的后门、图宽度参数以及加权可满足性问题等多个维度。尽管不同表述形式存在显著差异，我们运用经典复杂度理论的工具揭示了若干非平凡的关联。这引导我们构建了包含五个主要等价类的假说层次结构，其中若干核心结论包括：我们证明针对以有界路径宽度图、有界树宽度图或对数树深图为调制器参数化的SAT问题超越暴力搜索，实质上是同一问题，且等价于在深度为$\epsilon n$的电路上超越暴力搜索；我们证明针对强2-SAT后门超越暴力搜索，等价于针对对数路径宽度调制器超越暴力搜索；我们证明针对强Horn后门超越暴力搜索，等价于针对任意电路SAT问题超越暴力搜索。