We revisit the (structurally) parameterized complexity of Induced Matching and Acyclic Matching, two problems where we seek to find a maximum independent set of edges whose endpoints induce, respectively, a matching and a forest. Chaudhary and Zehavi~[WG '23] recently studied these problems parameterized by treewidth, denoted by $\mathrm{tw}$. We resolve several of the problems left open in their work and extend their results as follows: (i) for Acyclic Matching, Chaudhary and Zehavi gave an algorithm of running time $6^{\mathrm{tw}}n^{\mathcal{O}(1)}$ and a lower bound of $(3-\varepsilon)^{\mathrm{tw}}n^{\mathcal{O}(1)}$ (under the SETH); we close this gap by, on the one hand giving a more careful analysis of their algorithm showing that its complexity is actually $5^{\mathrm{tw}} n^{\mathcal{O}(1)}$, and on the other giving a pw-SETH-based lower bound showing that this running time cannot be improved (even for pathwidth), (ii) for Induced Matching we show that their $3^{\mathrm{tw}} n^{\mathcal{O}(1)}$ algorithm is optimal under the pw-SETH (in fact improving over this for pathwidth is \emph{equivalent} to falsifying the pw-SETH) by adapting a recent reduction for \textsc{Bounded Degree Vertex Deletion}, (iii) for both problems we give FPT algorithms with single-exponential dependence when parameterized by clique-width and in particular for \textsc{Induced Matching} our algorithm has running time $3^{\mathrm{cw}} n^{\mathcal{O}(1)}$, which is optimal under the pw-SETH from our previous result.

翻译：我们重新审视了诱导匹配与无环匹配这两个问题的（结构）参数化复杂度，在这两个问题中，我们试图找到一个最大的独立边集，其端点分别诱导出一个匹配和一个森林。Chaudhary 和 Zehavi~[WG '23] 最近研究了以树宽 $\mathrm{tw}$ 为参数时这些问题。我们解决了他们工作中遗留的几个问题，并将他们的结果扩展如下：(i) 对于无环匹配，Chaudhary 和 Zehavi 给出了一个运行时间为 $6^{\mathrm{tw}}n^{\mathcal{O}(1)}$ 的算法以及一个 $(3-\varepsilon)^{\mathrm{tw}}n^{\mathcal{O}(1)}$ 的下界（在 SETH 假设下）；我们通过更仔细地分析他们的算法，证明其复杂度实际上为 $5^{\mathrm{tw}} n^{\mathcal{O}(1)}$，并基于 pw-SETH 给出一个下界，表明该运行时间无法改进（即使对于路径宽也是如此），从而弥合了这一差距；(ii) 对于诱导匹配，我们通过调整最近用于 \textsc{Bounded Degree Vertex Deletion} 的归约，证明他们提出的 $3^{\mathrm{tw}} n^{\mathcal{O}(1)}$ 算法在 pw-SETH 假设下是最优的（事实上，对于路径宽改进该算法 \emph{等价于} 证伪 pw-SETH）；(iii) 对于这两个问题，我们给出了当以团宽为参数时具有单指数依赖性的 FPT 算法，特别是对于 \textsc{Induced Matching}，我们的算法运行时间为 $3^{\mathrm{cw}} n^{\mathcal{O}(1)}$，这根据我们之前的结果，在 pw-SETH 假设下是最优的。