The negative cycles polyhedron and hardness of checking some polyhedral properties

Given a graph G = (V, E) and a weight function on the edges w : E 7→ 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 [11], 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 [11] for non 0/1-polyhedra, and comes in contrast with the polynomiality of vertex enumeration for 0/1-polytopes [8]. 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 a polyhedron in Rn within a factor of 12/n.