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闭包是一个图中没有出边的顶点子集，即不存在从子集中顶点指向子集外顶点的边。闭包问题（英语：closure problem）是则是找到带权图中使得权之和最大或最小的子集。闭包问题可以看作最大流问题的简化版，在多项式时间内被解决。实际上，是否有环对于找到闭包没有影响。[26]

闭包是一个图中没有出边的顶点子集，即不存在从子集中顶点指向子集外顶点的边。闭包问题（英语：closure problem）是则是找到带权图中使得权之和最大或最小的子集。闭包问题可以看作最大流问题的简化版，在多项式时间内被解决。实际上，是否有环对于找到闭包没有影响。[26]

=== Path algorithms ===

=== Path algorithms 最短或最长路径问题 ===

Some algorithms become simpler when used on DAGs instead of general graphs, based on the principle of topological ordering. For example, it is possible to find [[shortest path]]s and [[longest path problem|longest paths]] from a given starting vertex in DAGs in linear time by [[Topological sorting#Application to shortest path finding|processing the vertices in a topological order]], and calculating the path length for each vertex to be the minimum or maximum length obtained via any of its incoming edges.<ref>Cormen et al. 2001, Section 24.2, Single-source shortest paths in directed acyclic graphs, pp. 592–595.</ref> In contrast, for arbitrary graphs the shortest path may require slower algorithms such as [[Dijkstra's algorithm]] or the [[Bellman–Ford algorithm]],<ref>Cormen et al. 2001, Sections 24.1, The Bellman–Ford algorithm, pp. 588–592, and 24.3, Dijkstra's algorithm, pp. 595–601.</ref> and longest paths in arbitrary graphs are [[NP-hard]] to find.<ref>Cormen et al. 2001, p. 966.</ref>

Some algorithms become simpler when used on DAGs instead of general graphs, based on the principle of topological ordering. For example, it is possible to find [[shortest path]]s and [[longest path problem|longest paths]] from a given starting vertex in DAGs in linear time by [[Topological sorting#Application to shortest path finding|processing the vertices in a topological order]], and calculating the path length for each vertex to be the minimum or maximum length obtained via any of its incoming edges.<ref>Cormen et al. 2001, Section 24.2, Single-source shortest paths in directed acyclic graphs, pp. 592–595.</ref> In contrast, for arbitrary graphs the shortest path may require slower algorithms such as [[Dijkstra's algorithm]] or the [[Bellman–Ford algorithm]],<ref>Cormen et al. 2001, Sections 24.1, The Bellman–Ford algorithm, pp. 588–592, and 24.3, Dijkstra's algorithm, pp. 595–601.</ref> and longest paths in arbitrary graphs are [[NP-hard]] to find.<ref>Cormen et al. 2001, p. 966.</ref>

Some algorithms become simpler when used on DAGs instead of general graphs, based on the principle of topological ordering. For example, it is possible to find shortest paths and longest paths from a given starting vertex in DAGs in linear time by processing the vertices in a topological order, and calculating the path length for each vertex to be the minimum or maximum length obtained via any of its incoming edges. In contrast, for arbitrary graphs the shortest path may require slower algorithms such as Dijkstra's algorithm or the Bellman–Ford algorithm, and longest paths in arbitrary graphs are NP-hard to find.

Some algorithms become simpler when used on DAGs instead of general graphs, based on the principle of topological ordering. For example, it is possible to find shortest paths and longest paths from a given starting vertex in DAGs in linear time by processing the vertices in a topological order, and calculating the path length for each vertex to be the minimum or maximum length obtained via any of its incoming edges. In contrast, for arbitrary graphs the shortest path may require slower algorithms such as Dijkstra's algorithm or the Bellman–Ford algorithm, and longest paths in arbitrary graphs are NP-hard to find.

基于拓扑排序原理，有些算法用在 DAGs 上代替一般图时变得更简单。例如，通过处理拓扑有序中的顶点，并计算每个顶点的路径长度为通过任一传入边得到的最小或最大长度，可以在线性时间内从给定的 DAGs 顶点找到最短路径和最长路径。相反，对于任意图，最短路径可能需要较慢的算法，如 Dijkstra 算法或 Bellman-Ford 算法，而任意图中的最长路径是 np 难以找到的。

 

 

的贝尔曼-福特算法等。[28]最长路径则是一个NP困难问题。[29]

== Computational problems ==

== Computational problems ==