电路与后门：SETH的五种形态
Circuits and Backdoors: Five Shades of the SETH
AI总结：
本文将强指数时间假设（SETH）的多种合理弱化形式整合为五个稳固性递增的等价类层级结构，并通过独立集等非SAT问题的特征示例展示了该框架在结构性分类基于SETH下界方面的适用性。
摘要：
强指数时间假设（SETH）是（细粒度）参数化复杂性中的一个标准假设，许多紧致下界都是基于它建立的。我们考虑了对 SETH 的若干合理弱化，其来源包括 电路复杂性、SAT 求解的后门、图宽度参数以及加权可满足性问题。我们的目标是得出既作为假设更为合理，又能捕捉有趣且稳健的复杂性概念的表述。利用经典复杂性理论中的若干工具，我们能够将这些众多假设整合为五个主要等价类的层级结构，其稳固性逐渐递增。这一框架作为迈向结构性分类的一步，旨在将各种基于 SETH 的下界划分到中间等价类中。为了说明我们框架的适用性，对于每个等价类，我们给出了至少一个与该类等价的（非 SAT）问题，作为特征性示例应用。作为我们的主要展示，我们考虑了 Independent Set 的一种自然参数化，即通过到若干标准图类的顶点删除距离进行参数化。我们提供了对打破此类界限难度的精确刻画，特别地，证明了对于 Cograph+kv 或 Block+kv 图获得 $(2-\varepsilon)^kn^{O(1)}$ 时间算法，等价于为深度为 $\varepsilon n$ 的电路获得快速可满足性算法；而解决 Interval+kv 图的加权版本，则等价于为以 2-SAT 后门参数化的 SAT 获得快速可满足性算法这一（看似）更难的问题。
英文摘要：
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.
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