Discrete & Computational Geometry

, Volume 39, Issue 1–3, pp 174–190 | Cite as

Generating All Vertices of a Polyhedron Is Hard

  • Leonid Khachiyan
  • Endre Boros
  • Konrad Borys
  • Khaled Elbassioni
  • Vladimir Gurvich
Open Access


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 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. Equiva lently, the complexity of generating vertices and extreme rays of polyhedra remains open.


Discrete Comput Geom Conjunctive Normal Form Satisfying Assignment Negative Cycle Vertex Enumeration 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Leonid Khachiyan
  • Endre Boros
    • 1
  • Konrad Borys
    • 1
  • Khaled Elbassioni
    • 2
  • Vladimir Gurvich
    • 1
  1. 1.RUTCORRutgers UniversityPiscatawayUSA
  2. 2.Department 1Max-Planck-Institut für InformatikSaarbrückenGermany

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