Characterization of the vertices and extreme directions of the negative cycle polyhedron and harness 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 no output polynomial-time algorithm to generate all the vertices of a 0/1-polyhedron. This strengthens the NP-hardness result of [2] for non 0/1 polyhedra, and comes in contrast with the polynomiality of vertex enumeration for 0/1-polytopes [1]. 1 The polyhedron of negative weighted-flows Given a directed graph G = (V,E) and a weight function w : E → R on its arcs, consider the following polyhedron: