原始路径宽度强指数时间假设（Primal Pathwidth SETH）
The Primal Pathwidth SETH
AI总结：
本文提出原始路径宽度-强指数时间假设（pw-SETH），证明参数化复杂性中多个细粒度下界问题均与该假设等价，既给出更精确的下界，也提升了既有下界的可信度。
摘要：
受动态规划（DP）在参数化复杂性中重要性的启发，我们研究了若干细粒度问题，例如以下示例：（i）支配集（Dominating Set）问题能否在$(3-ε)^{pw}n^{O(1)}$时间内求解？（其中$pw$为路径宽度）（ii）着色（Coloring）问题能否在$pw^{(1-ε)pw}n^{O(1)}$时间内求解？（iii）能否在$n^{(1-ε)k}$时间内找到两个大小为$k$的独立集之间的短重构路径？这类问题已得到广泛研究：部分问题在强指数时间假设（SETH）下答案是否定的，另一部分则在指数时间假设（ETH）下有粗粒度下界。尽管上述问题看似“本质上等价”——它们都质疑简单DP是否存在改进空间，但所涉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。\n本文的主要贡献在于证明，尽管复杂度各异，这些问题不仅本质上等价，实际上是同一问题的不同表现形式。我们通过提出一个自然的复杂性假设实现这一点，将其称为原始路径宽度-强指数时间假设（Primal Pathwidth-Strong Exponential Time Hypothesis，pw-SETH），该假设指出：对于任意$ε>0$，3-SAT问题无法在$(2-ε)^{pw}n^{O(1)}$时间内求解，其中$pw$是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（包括上述示例）均与pw-SETH等价，因此彼此之间也等价。这一结论一方面能为那些此前下界未确定指数中常数项的问题，给出精确的细粒度下界；另一方面也提升了此前在SETH下得到的下界的可信度——因为我们证明，突破其中任意一个下界都意味着要突破所有（既有及新增的）下界；同时我们还证明pw-SETH比SETH更具合理性。
英文摘要：
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 $(3-ε)^{pw}n^{O(1)}$? (where $pw$ is the pathwidth) (ii) can Coloring be solved in time $pw^{(1-ε)pw}n^{O(1)}$? (iii) can a short reconfiguration between two size-$k$ independent sets be found in time $n^{(1-ε)k}$? 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 $(2-ε)^{pw}n^{O(1)}$, for any $ε>0$, 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.
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