Generating All Vertices of a Polyhedron Is Hard

Leonid Khachiyan, Endre Boros, Konrad Borys, Khaled Elbassioni, Vladimir Gurvich
2008 Discrete & Computational Geometry  
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
more » ... ting problems from linear programming: (i) Given an infeasible system of linear inequalities, generating all minimal infeasible subsystems is hard. Yet, for generating 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 the case of bounded polyhedra the complexity remains open. Equivalently the complexity of generating vertices and extreme rays of polyhedra remains open.
doi:10.1007/s00454-008-9050-5 fatcat:s5been3h6bcnthbkmvkbhks3m4