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The negative cycles polyhedron and hardness of checking some polyhedral properties

Author

Listed:
  • Endre Boros
  • Khaled Elbassioni
  • Vladimir Gurvich
  • Hans Tiwary

Abstract

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. Copyright Springer Science+Business Media, LLC 2011

Suggested Citation

  • Endre Boros & Khaled Elbassioni & Vladimir Gurvich & Hans Tiwary, 2011. "The negative cycles polyhedron and hardness of checking some polyhedral properties," Annals of Operations Research, Springer, vol. 188(1), pages 63-76, August.
    • Handle: RePEc:spr:annopr:v:188:y:2011:i:1:p:63-76:10.1007/s10479-010-0690-5
    DOI: 10.1007/s10479-010-0690-5
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    Cited by:

    1. Gustavo Angulo & Shabbir Ahmed & Santanu S. Dey & Volker Kaibel, 2015. "Forbidden Vertices," Mathematics of Operations Research, INFORMS, vol. 40(2), pages 350-360, February.

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