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. To illustrate the applicability of our framework, for each of our classes we give at least one (non-SAT) problem which is equivalent to the class as a characteristic example application. As our main showcase, we consider a natural parameterization of Independent Set by vertex deletion distance from several standard graph classes. We provide precise characterizations of the difficulty of breaking such bounds, in particular proving that obtaining $(2-\varepsilon)^kn^{O(1)}$ time algorithms for Cograph$+kv$ or Block$+kv$ graphs is equivalent to obtaining a fast satisfiability algorithm for circuits of depth $\varepsilon n$; while solving the weighted version for Interval$+kv$ graphs is equivalent to the (seemingly) harder problem of obtaining a fast satisfiability algorithm for SAT parameterized by a 2-SAT backdoor.

翻译：强指数时间假设（SETH）是（细粒度）参数化复杂性理论中的标准假设，许多紧下界都基于该假设。我们考虑了一系列合理的SETH弱化形式，其来源包括：（i）电路复杂性（ii）SAT求解的后门（iii）图宽度参数（iv）加权可满足性问题。我们的目标是提出既作为假设更具合理性，又能捕捉有趣且稳健的复杂性概念的表述形式。借助经典复杂性理论中的多种工具，我们将这些众多假设整合为五个主要等价类的层次结构，其稳健性逐级增强。该框架为将各类基于SETH的下界结构性地分类到中间等价类提供了基础。为说明框架的适用性，我们为每个等价类提供了至少一个（非SAT）问题作为特征性示例应用。作为主要展示案例，我们研究了通过顶点删除距离参数化独立集问题到若干标准图类的自然参数化形式。我们精确刻画了突破此类下界的难度，特别证明了针对Cograph$+kv$或Block$+kv$图获得$(2-\varepsilon)^kn^{O(1)}$时间算法，等价于获得深度为$\varepsilon n$的电路的快速可满足性算法；而求解Interval$+kv$图的加权版本则等价于（看似）更难的问题——获得针对以2-SAT后门参数化的SAT的快速可满足性算法。