# Editing Shortest path

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'''Corollary''': Assuming our graphs have no cycles of negative weight, the restriction that the shortest paths be simple is immaterial. Therefore, we will assume in the foregoing discussion that our graphs have no cycles of negative weight, for the problem of finding shortest ''simple'' paths in graphs containing negative-weight cycles is NP-complete. | '''Corollary''': Assuming our graphs have no cycles of negative weight, the restriction that the shortest paths be simple is immaterial. Therefore, we will assume in the foregoing discussion that our graphs have no cycles of negative weight, for the problem of finding shortest ''simple'' paths in graphs containing negative-weight cycles is NP-complete. | ||

− | '''Corollary''': In a finite | + | '''Corollary''': In a finite graph, a shortest path always exists. (To prove this we simply use the fact that the graph has a finite number of simple paths, and only simple paths need be considered. So one of them must be the shortest.) |

==Variations== | ==Variations== |