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 (PP-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 PP-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 PP-SETH is more plausible than the SETH.

翻译：受动态规划在参数化复杂度中重要性的启发，我们考虑若干精细粒度问题，例如以下实例：(i) 支配集能否在时间$(3-\epsilon)^{pw}n^{O(1)}$内求解？（其中$pw$为路径宽度）(ii) 着色问题能否在时间$pw^{(1-\epsilon)pw}n^{O(1)}$内求解？(iii) 两个大小为$k$的独立集之间的短重配置能否在时间$n^{(1-\epsilon)k}$内找到？此类问题已有深入研究：在某些情形下，基于SETH的答案为否定；而在其他情形下，基于ETH已知粗粒度下界。尽管上述问题看似“道德等价”——均询问简单动态规划能否改进——但所涉问题的时间复杂度却差异悬殊，范围涵盖从单指数FPT到XNLP完全问题。本文的主要贡献在于证明：尽管复杂度各异，这些问题不仅道德等价，实则本质相同。为此，我们提出一个自然的复杂度假设——原始路径宽度-强指数时间假说（PP-SETH），该假设断言：对于任意$\epsilon>0$，3-SAT无法在时间$(2-\epsilon)^{pw}n^{O(1)}$内求解，其中$pw$为输入原始图的路径宽度。我们进而证明参数化复杂度中的大量精细粒度问题（包括上述实例）均与PP-SETH等价，从而彼此等价。这使我们既能获得此前下界未确定指数常数的尖锐精细粒度下界，也能增强对先前基于SETH已知下界的信心——因为证明打破任一此类下界需同时打破所有（新旧）下界；同时我们亦表明PP-SETH比SETH更为合理。