In this paper, we design a framework to obtain efficient algorithms for several problems with a global constraint (acyclicity or connectivity) such as Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree, Longest Induced Path, and Feedback Vertex Set. We design a meta-algorithm that solves all these problems and whose running time is upper bounded by $2^{O(k)}\cdot n^{O(1)}$, $2^{O(k \log(k))}\cdot n^{O(1)}$, $2^{O(k^2)}\cdot n^{O(1)}$ and $n^{O(k)}$ where $k$ is respectively the clique-width, $\mathbf{Q}$-rank-width, rank-width and maximum induced matching width of a given decomposition. Our approach simplifies and unifies the known algorithms for each of the parameters and its running time matches asymptotically also the running times of the best known algorithms for basic NP-hard problems such as Vertex Cover and Dominating Set. Our framework is based on the $d$-neighbor equivalence defined in [Bui-Xuan, Telle and Vatshelle, TCS 2013] and the rank-based approach introduced in [Bodlaender, Cygan, Kratsch and Nederlof, ICALP 2013]. The results we obtain highlight the importance of the $d$-neighbor equivalence relation on the algorithmic applications of width measures. We also prove that our framework could be useful for $W[1]$-hard problems parameterized by clique-width such as Max Cut and Maximum Minimal Cut. For these latter problems, we obtain $n^{O(k)}$, $n^{O(k)}$ and $n^{2^{O(k)}}$ time algorithms where $k$ is respectively the clique-width, the $\mathbf{Q}$-rank-width and the rank-width of the input graph.

翻译：本文设计了一个统一框架，用于高效解决若干具有全局约束（无环性或连通性）的问题，包括连通支配集、带权顶点斯坦纳树、最大导出树、最长导出路径及反馈顶点集。我们提出了一种元算法，可同时求解上述所有问题，其运行时间上界分别为 $2^{O(k)}\cdot n^{O(1)}$、$2^{O(k \log(k))}\cdot n^{O(1)}$、$2^{O(k^2)}\cdot n^{O(1)}$ 和 $n^{O(k)}$，其中 $k$ 分别表示给定分解的团宽度、$\mathbf{Q}$-秩宽度、秩宽度及最大导出匹配宽度。该方法简化并统一了针对各参数的已知算法，其渐近运行时间也与顶点覆盖、支配集等经典NP难问题已知最优算法的时间复杂度相匹配。本框架基于 [Bui-Xuan, Telle and Vatshelle, TCS 2013] 提出的 $d$-邻域等价关系，以及 [Bodlaender, Cygan, Kratsch and Nederlof, ICALP 2013] 引入的秩基方法。我们的结果凸显了 $d$-邻域等价关系在宽度测度算法应用中的重要性。同时证明，该框架可有效处理以团宽度为参数的 $W[1]$-难问题，如最大割与最大最小割。针对后者，我们分别给出了运行时间为 $n^{O(k)}$、$n^{O(k)}$ 和 $n^{2^{O(k)}}$ 的算法，其中 $k$ 依次表示输入图的团宽度、$\mathbf{Q}$-秩宽度和秩宽度。