Motivated by the importance of dynamic programming (DP) in parameterized complexity, we consider several fine-grained questions, such as the following examples: (i) can Dominating Set be solved in time $(3-\epsilon)^{pw}n^{O(1)}$? (where $pw$ is the pathwidth) (ii) can Coloring be solved in time $pw^{(1-\epsilon)pw}n^{O(1)}$? (iii) can a short reconfiguration between two size-$k$ independent sets be found in time $n^{(1-\epsilon)k}$? Such questions are well-studied: in some cases the answer is No under the SETH, while in others coarse-grained lower bounds are known under the ETH. Even though questions such as the above seem "morally equivalent" as they all ask if a simple DP can be improved, the problems concerned have wildly varying time complexities, ranging from single-exponential FPT to XNLP-complete. This paper's main contribution is to show that, despite their varying complexities, these questions are not just morally equivalent, but in fact they are the same question in disguise. We achieve this by putting forth a natural complexity assumption which we call the Primal Pathwidth-Strong Exponential Time Hypothesis (pw-SETH) and which states that 3-SAT cannot be solved in time $(2-\epsilon)^{pw}n^{O(1)}$, for any $\epsilon>0$, where $pw$ is the pathwidth of the primal graph of the input. We then show that numerous fine-grained questions in parameterized complexity, including the ones above, are equivalent to the pw-SETH, and hence to each other. This allows us to obtain sharp fine-grained lower bounds for problems for which previous lower bounds left a constant in the exponent undetermined, but also to increase our confidence in bounds which were previously known under the SETH, because we show that breaking any one such bound requires breaking all (old and new) bounds; and because we show that the pw-SETH is more plausible than the SETH.

翻译：受动态规划（DP）在参数化复杂性中的重要性驱动，我们考虑了若干细粒度问题，例如以下示例：（i）支配集问题能否在$(3-\epsilon)^{pw}n^{O(1)}$时间内求解？（其中$pw$为路径宽度）（ii）着色问题能否在$pw^{(1-\epsilon)pw}n^{O(1)}$时间内求解？（iii）能否在$n^{(1-\epsilon)k}$时间内找到两个规模为$k$的独立集之间的短重构序列？此类问题已被广泛研究：在某些情况下，根据强指数时间假设（SETH）答案为否；而在其他情况下，根据指数时间假设（ETH）已知粗粒度下界。尽管上述问题看似"道德等价"，因为它们都询问简单DP能否改进，但所涉及问题的时间复杂度差异极大，从单指数固定参数可解（FPT）到XNLP完全问题不等。本文的主要贡献在于证明，尽管这些问题复杂度各异，但它们不仅是道德等价的，实际上更是同一问题的不同表现形式。我们通过提出一个称为原始路径宽度强指数时间假设（pw-SETH）的自然复杂性假设来实现这一目标，该假设断言：对于任意$\epsilon>0$，无法在$(2-\epsilon)^{pw}n^{O(1)}$时间内求解3-SAT问题，其中$pw$为输入原始图的路径宽度。随后我们证明，参数化复杂性中的众多细粒度问题（包括上述问题）均等价于pw-SETH，从而彼此等价。这使得我们能够为那些先前下界未确定指数中常数的问題获得精确的细粒度下界，同时也增强了我们对基于SETH已知界限的信心——因为我们证明突破任一界限将导致所有（新旧）界限被同时突破；并且我们表明pw-SETH比SETH更具合理性。