Characterization of the Vertices and Extreme Directions of the Negative Cycles Polyhedron and Hardness of Generating Vertices of 0/1-Polyhedra

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). As a corollary, 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 [9] for non 0/1-polyhedra, and comes in contrast with the polynomiality of vertex enumeration for 0/1-polytopes [7].