Extended Formulations for Radial Cones

Abstract: This paper studies extended formulations for radial cones at vertices of polyhedra, where the radial cone of a polyhedron $ P $ at a vertex $ v \in P $ is the polyhedron defined by the constraints of $ P $ that are active at $ v $. Given an extended formulation for $ P $, it is easy to obtain an extended formulation of comparable size for each its radial cones. On the contrary, it is possible that radial cones of $ P $ admit much smaller extended formulations than $ P $ itself. A prominent example of this type is the perfect-matching polytope, which cannot be described by subexponential-size extended formulations (Rothvo\ss{} 2014). However, Ventura & Eisenbrand (2003) showed that its radial cones can be described by polynomial-size extended formulations. Moreover, they generalized their construction to $ V $-join polyhedra. In the same paper, the authors asked whether the same holds for the odd-cut polyhedron, the blocker of the $ V $-join polyhedron. We answer this question negatively. Precisely, we show that radial cones of odd-cut polyhedra cannot be described by subexponential-size extended formulations. To obtain our result, for a polyhedron $ P $ of blocking type, we establish a general relationship between its radial cones and certain faces of the blocker of $ P $.