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In the video, the author discusses the longest path problem in directed graphs. He first shows how to find the longest path to each vertex in a directed acyclic graph (DAG), and then demonstrates how to compute the longest path to each vertex in a graph with cycles. The video makes it clear that the longest path problem is difficult to solve, but provides some helpful tips for those who want to try.

**00:00:00**In this video, the author discusses the problem of finding the longest path in a directed acyclic graph (DAG). He shows how to find the longest path to each vertex in the DAG, and how to compute the longest path to each vertex if we know the path to the initial vertex. Finally, he demonstrates how to solve the longest path problem by finding the longest path to each vertex.**00:05:00**The video discusses how to compute the longest path in a graph. The algorithm starts by sorting the vertices in topological order, and then computing the indegree of each vertex. Next, the longest path is computed as the maximum of the indegrees of the vertices that are already known to have a longest path.**00:10:00**The video discusses the longest path problem, which is a problem in graph theory involving computing the shortest duration that a path needs to complete. The longest path can be computed in an overlapping way with the topological sort, and it makes sense even in graphs with cycles.**00:15:00**The longest path problem in directed graphs is difficult to solve, compared to the simpler problem in directed graphs without cycles.

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