The canonical tree-decomposition theorem, given by Robertson and Seymour in their seminal graph minors series, turns out to be one of the most important tool in structural and algorithmic graph theory. In this paper, we provide the canonical tree decomposition theorem for digraphs. More precisely, we construct directed tree-decompositions of digraphs that distinguish all their tangles of order $k$, for any fixed integer $k$, in polynomial time. As an application of this canonical tree-decomposition theorem, we provide the following result for the directed disjoint paths problem: For every fixed $k$ there is a polynomial-time algorithm which, on input $G$, and source and terminal vertices $(s_1, t_1), \dots, (s_k, t_k)$, either 1. determines that there is no set of pairwise vertex-disjoint paths connecting each source $s_i$ to its terminal $t_i$, or 2.finds a half-integral solution, i.e., outputs paths $P_1, \dots, P_k$ such that $P_i$ links $s_i$ to $t_i$, so that every vertex of the graph is contained in at most two paths. Given known hardness results for the directed disjoint paths problem, our result cannot be improved for general digraphs, neither to fixed-parameter tractability nor to fully vertex-disjoint directed paths. As far as we are aware, this is the first time to obtain a tractable result for the $k$-disjoint paths problem for general digraphs. We expect more applications of our canonical tree-decomposition for directed results.

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规范有向树分解及其在有向不相交路径问题中的应用

Robertson 和 Seymour 在他们开创性的图未成年人系列中给出的规范树分解定理被证明是结构和算法图论中最重要的工具之一。在本文中，我们提供了有向图的规范树分解定理。更准确地说，我们构造了有向图的有向树分解，这些有向图在多项式时间内区分了所有 $k$ 阶缠结，对于任何固定整数 $k$。作为此规范树分解定理的应用，我们为有向不相交路径问题提供以下结果：对于每个固定的 $k$，存在多项式时间算法，该算法在输入 $G$ 上，以及源和终端顶点 $ (s_1, t_1), \dots, (s_k, t_k)$, 要么 1. 确定没有一组成对的顶点不相交路径将每个源 $s_i$ 连接到其终端 $t_i$，或 2. 找到半积分解，即输出路径 $P_1, \dots, P_k$ 使得 $P_i$ 将 $s_i$ 链接到 $t_i$，使得图的每个顶点最多包含在两条路径中. 给定有向不相交路径问题的已知硬度结果，对于一般有向图，无论是固定参数的易处理性还是完全顶点不相交的有向路径，我们的结果都无法改进。据我们所知，这是第一次为一般有向图的 $k$-disjoint 路径问题获得易于处理的结果。我们期望我们的规范树分解有更多应用来获得定向结果。对于一般有向图，无论是固定参数的易处理性还是完全顶点不相交的有向路径，我们的结果都无法改进。据我们所知，这是第一次为一般有向图的 $k$-disjoint 路径问题获得易于处理的结果。我们期望我们的规范树分解有更多应用来获得定向结果。对于一般有向图，无论是固定参数的易处理性还是完全顶点不相交的有向路径，我们的结果都无法改进。据我们所知，这是第一次为一般有向图的 $k$-disjoint 路径问题获得易于处理的结果。我们期望我们的规范树分解有更多应用来获得定向结果。