Annals of Operations Research

, Volume 188, Issue 1, pp 63–76 | Cite as

The negative cycles polyhedron and hardness of checking some polyhedral properties

  • Endre Boros
  • Khaled Elbassioni
  • Vladimir Gurvich
  • Hans Raj Tiwary


Given a graph G=(V,E) and a weight function on the edges w:E→ℝ, we consider the polyhedron P(G,w) of negative-weight flows on G, and get a complete characterization of the vertices and extreme directions of P(G,w). Based on this characterization, and using a construction developed in Khachiyan et al. (Discrete Comput. Geom. 39(1–3):174–190, 2008), we show that, unless P=NP, there is no output polynomial-time algorithm to generate all the vertices of a 0/1-polyhedron. This strengthens the NP-hardness result of Khachiyan et al. (Discrete Comput. Geom. 39(1–3):174–190, 2008) for non 0/1-polyhedra, and comes in contrast with the polynomiality of vertex enumeration for 0/1-polytopes (Bussiech and Lübbecke in Comput. Geom., Theory Appl. 11(2):103–109, 1998). As further applications, we show that it is NP-hard to check if a given integral polyhedron is 0/1, or if a given polyhedron is half-integral. Finally, we also show that it is NP-hard to approximate the maximum support of a vertex of a polyhedron in ℝ n within a factor of 12/n.


Flow polytope 0/1-polyhedron Vertex Extreme direction Enumeration problem Negative cycles Directed graph Half-integral polyhedra Maximum support Hardness of approximation 


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Endre Boros
    • 1
  • Khaled Elbassioni
    • 2
  • Vladimir Gurvich
    • 1
  • Hans Raj Tiwary
    • 3
  1. 1.RUTCORRutgers UniversityPiscatawayUSA
  2. 2.Max-Planck-Institut für InformatikSaarbrückenGermany
  3. 3.Technische Universität Berlin Fakultät II: Institut für MathematikBerlinGermany

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