DIMACS TR: 2005-21

Generating all vertices of a polyhedron is hard

Authors: Leonid Khachiyan, Endre Boros, Konrad Borys, Khaled Elbassioni and Vladimir Gurvich


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 (directed) negative cycles, it is an NP-complete 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 generating 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 keywords polytope, polyhedron, polytope-polyhedron problem, vertex, face, facet, enumeration problem, vertex enumeration, facet enumeration, graph, cycle, negative cycle, linear inequalities, feasible system

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