The negative cycles polyhedron and hardness of checking some polyhedral properties

Endre Boros (), Khaled Elbassioni (), Vladimir Gurvich () and Hans Tiwary ()

Annals of Operations Research, 2011, vol. 188, issue 1, 63-76

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

Keywords: Flow polytope; 0/1-polyhedron; Vertex; Extreme direction; Enumeration problem; Negative cycles; Directed graph; Half-integral polyhedra; Maximum support; Hardness of approximation (search for similar items in EconPapers)
Date: 2011
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10479-010-0690-5

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