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Generating All Vertices of a Polyhedron Is Hard

2007
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Discrete & Computational Geometry
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We show that generating all negative cycles of a weighted graph is a hard enumeration problem, in both the directed and undirected cases. More precisely, given a family of negative (directed) cycles, it is an NPcomplete problem to decide whether this family can be extended or there are no other negative (directed) cycles in the graph, implying that (directed) negative cycles cannot be generated in polynomial output time, unless P=NP. As a corollary, we solve in the negative two well-known

doi:10.1007/s00454-006-1259-6
fatcat:4oausme7ejfolhbegnti222y74