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Generating all vertices of a polyhedron is hard.

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u‰‰ŽÒFDr.@Vladimir A.Gurvich
@@@@@ (Rutgers Center for Operations Reseach,
@@@@@ The State University of New Jersey,
@@@@@ Adjunct Professor and@Researcher )

ƒ^ƒCƒgƒ‹FGenerating all vertices of a polyhedron is hard.


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We show that generating all negative directed cycles of a directed graph is a hard generating problem. More precisely, given a digraph and a collection of its negative dicycles, it is NP-hard to decide whether this collection is complete.
As a corollary we solve in negative two well-known generating problems from linear programming:

(i) Given an infeasible system of linear inequalities, generating all its minimal infeasible subsystems(so-called Helly subsystems) is hard.
@ Yet, for maximal feasible subsystems the complexity remains open.

(ii) Given a feasible system of linear inequalities, generating all vertices of the corresponding polyhedron@is hard. Yet, in case of bounded polyhedra the complexity remains open

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